0$. Note that \eqref{mc} is a particular case of such metric. Let $\cI\subset\cun(\oX)$ be the principal ideal generated by the function $x$. Recall \cite{meni96c} that a \emph{cusp vector field} is a smooth vector field $V$ on $\oX$ such that $dx(V)\in\cI^2$. The space of cusp vector fields forms a Lie subalgebra $\ccV$ of the Lie algebra $\cV$ of smooth vector fields on $\oX$. In fact, there exists a natural vector bundle $\ctX$ over $\oX$ whose space of smooth sections is $\ccV$, and a natural map $\ctX\to TX$ which induces the inclusion $\ccV\hookrightarrow \cV$. Let $E,F\to\oX$ be smooth vector bundles. The space of cusp differential operators $\Diffc(\oX,E,F)$ is the space of those differential operators which in local trivializations can be written as composition of cusp vector fields and smooth bundle morphisms down to $x=0$. The \emph{normal operator} of $P\in\Diffc(\oX,E,F)$ is the family of operators defined by \[\rz\ni\xi\mapsto\cN(P)(\xi):= \left(e^{i\xi/x}Pe^{-i\xi/x}\right)_{|x=0}\in\Diff(M,E_{|M},F_{|M}).\] \begin{example}\label{ex1} $\cN(x^2\px)(\xi)=i\xi$. \end{example} Note that $\ker \cN=\cI\cdot\Diffc$, which we denote again by $\cI$. The normal operator map is linear and multiplicative. It is also invariant under the conjugation by powers of $x$. Namely, if $P\in\Diffc$ and $s\in\cz$ then $x^s Px^{-s}\in\Diffc$ and $\cN(x^s Px^{-s})=\cN(P)$. Concerning taking the (formal) adjoint, one needs to specify the volume form on the boundary. \begin{lemma} \label{lema6} Let $P\in\Diffc(\oX,E,F)$ be a cusp operator and $P^*$ its adjoint with respect to $g_0$. Then $\cN(P^*)(\xi)$ is the adjoint of $\cN(P)(\xi)$ with respect to the metric on $E_{|M},F_{|M}$ induced by restriction, and the volume form ${a_{0}}^{1/2} \vol_{h_0}$ (see Eq.\ \eqref{cume2}). \end{lemma} The principal symbol of a cusp operator on $X$ extends as a map on the cusp cotangent bundle down to $x=0$. This implies that a cusp operator of positive order cannot be elliptic at $x=0$ in the usual sense. A cusp operator is called cusp-elliptic if its principal symbol is invertible on $\ctsX\setminus\{0\}$ down to $x=0$. \begin{definition} A cusp operator is called \emph{fully elliptic} if it is cusp-elliptic and if its normal operator is invertible for all values of $\xi\in\rz$. \end{definition} An operator $H\in x^{-l}\Diffc^k(\oX,E,F)$ is called a cusp differential operator of type $(k,l)$. Fix a product decomposition of $X$ near $M$, compatible with the boundary-defining function $x$. This gives a splitting of the cusp cotangent bundle on $\oX$ in a neighborhood of $M$: \begin{equation}\label{decom} \ctsX\simeq T^*M\oplus\langle x^{-2}dx\rangle. \end{equation} \begin{lemma}\label{prop3} The de Rham differential $d:\cun(X)\to \cun(X,T^*X)$ restricts to a cusp differential operator $d:\cun(\oX)\to\cun(\oX,\ctsX)$. Its normal operator in the decomposition \eqref{decom} is \[\cN(d)(\xi)= \begin{bmatrix}d^M\\i\xi\end{bmatrix}\] where $d^M$ is the partial de Rham differential in the $M$ factor of the product decomposition. \end{lemma} \proof Let $\omega\in\cun(\oX)$ and decompose $d\omega$ according to \eqref{decom}: \[d\omega=d^M\omega + \px(\omega) dx=d^M\omega+x^2\px(\omega)\frac{dx}{x^2}.\] Since $d^M$ commutes with $x$, it follows from the definition that $\cN(d^M)=d^M$. The result follows using Example \ref{ex1}. \qed \subsection{Relative de Rham cohomology}\label{rdR} Recall that the cohomology of $\oX$ and the relative cohomology groups of $(\oX,M)$ (with real coefficients) can be computed using smooth forms as follows: let $\Lambda^*(\oX)$ denote the space of forms smooth on $\oX$ down to the boundary. Let $\Lambda^*(\oX,M)$ denote the subspace of those forms whose pull-back to $M$ vanishes. These spaces form complexes for the de Rham differential (because $d$ commutes with pull-back to $M$) and their quotient is the de Rham complex of $M$: \[0\to \Lambda^*(\oX,M)\hookrightarrow \Lambda^*(\oX)\to\Lambda^*(M)\to 0.\] The induced long exact sequence in cohomology is just the long exact sequence of the pair $(\oX,M)$. \subsection{Cusp de Rham cohomology} Notice that $d$ preserves the space of cusp differential forms. Indeed, since $d$ is a derivation and using Lemma \ref{prop3}, it suffices to check this property for a set of local generators of $\cun(\oX,\ctsX)$. Choose local coordinates $(y_j)$ on $M$ and take as generators $x^{-2}dx$ and $dy_j$, which are closed. Let $\cH^*(X)$ denote the cohomology of the complex of cusp forms $(\cun(\oX,\Lambda^*(\ctX)), d)$ with respect to the de Rham differential. \begin{proposition}\label{dec} $\cH^k(X)=H^k(X)\oplus H^{k-1}(M)^2$. \end{proposition} \proof The short exact sequence of de Rham complexes \begin{equation*} 0\to \Lambda^*(T\oX)\hookrightarrow \Lambda^*(\ctX)\to \Lambda^{*-1}(TM)^2\to 0, \end{equation*} where the second map is given by \begin{equation}\label{restfr} \Lambda^k(\ctX)\ni\omega\mapsto \left((x^2\px\lrcorner \omega)_{x=0}, (\partial_x(x^2\px\lrcorner \omega))_{x=0}\right), \end{equation} gives rise to a long exact sequence in cohomology. Now the composition \[\Lambda^*(T\oX)\hookrightarrow \Lambda^*(\ctX)\hookrightarrow \Lambda^*(TX)\] is a quasi-isomorphism, since de Rham cohomology can be computed either with smooth forms on $\oX$, or with smooth forms on $X$. Thus in cohomology the map induced from $\Lambda^*(T\oX)\hookrightarrow \Lambda^*(\ctX)$ is injective. \qed \section{The magnetic Laplacian}\label{sectionmagn} \subsection{The magnetic Laplacian on a Riemannian manifold} A \emph{magnetic field} $B$ on the Riemannian manifold $(X,g)$ is an exact real-valued $2$-form. A \emph{vector potential} $A$ associated to $B$ is a $1$-form such that $dA=B$. We form the \emph{magnetic Laplacian} acting on $\cun(X)$: \[\Delta_A:=\dA^*\dA.\] This formula makes sense for complex-valued $1$-forms $A$. Note that when $A$ is real, $\dA$ is a metric connection on the trivial bundle $\underline{\cz}$ with the canonical metric, and $\Delta_A$ is the connection Laplacian. If we alter $A$ by adding to it a real exact form, say $A'=A+df$, the resulting magnetic Laplacian satisfies \[\Delta_{A'}=e^{-if}\Delta_Ae^{if}\] so it is unitarily equivalent to $\Delta_A$ in $L^2(X,g)$. Therefore if $H^1_{\mathrm{dR}}(X)=0$ (for instance if $\pi_1(X)$ is finite; see \cite{gh}) then $\Delta_A$ depends, up to unitary equivalence, only on the magnetic field $B$. This property is called \emph{gauge invariance}. One usually encounters gauge invariance as a consequence of $1$-connectedness (i.e., $\pi_1=0$). But in dimensions at least $4$, every finitely presented group (in particular, every finite group) can be realized as $\pi_1$ of a compact manifold. Thus the hypothesis $\pi_1=0$ is unnecessarily restrictive, it is enough to assume that its abelianisation is finite. While the properties of $\Delta_A$ in $\rz^n$ with the flat metric are quite well understood, the (absence of) essential spectrum of magnetic Laplacians on other manifolds has not been much studied so far. One exception is the case of bounded geometry, studied in \cite{KS}. However our manifolds are not of bounded geometry because the injectivity radius tends to $0$ at infinity. \subsection{Magnetic fields and cohomology}\label{s:mfc} Recall that $A$ is a (smooth) \emph{cusp $1$-form} (not to be confused with the notion of cusp form from automorphic form theory) on $\oX$ if $A\in\cun(X,T^*X)$ is a real-valued $1$-form satisfying near $\partial X$ \begin{equation}\label{aft} A=\varphi(x)\frac{dx}{x^2}+\theta(x) \end{equation} where $\varphi\in\cun(\oX)$ and $\theta\in\cun([0,\varepsilon)\times M, \Lambda^1(M))$, or equivalently $A$ is a smooth section in $\ctsX$ over $\oX$. \begin{proposition}\label{p:cuspduB} Let $B$ be a cusp $2$-form. Suppose that $B$ is exact on $X$, and its image by the map \eqref{restfr} is exact on $M$. Then there exists a smooth cusp $1$-form $A$ on $\oX$ such that $dA=B$. \end{proposition} \proof Note that $B$ is exact as a form on $X$, so $dB=0$ on $X$. By continuity, $dB=0$ on $\oX$ (in the sense of cusp forms) so $B$ defines a cusp cohomology $2$-class. By hypothesis, this class maps to $0$ by restriction to $X$. Now the pull-back of $B$ to the level surfaces $\{x=\varepsilon\}$ is closed; by continuity, the image of $B$ through the map \eqref{restfr} is closed on $M$. Assuming that this image is exact, it follows from Proposition \ref{dec} that $B$ is exact as a cusp form. \qed By Lemmata \ref{lema6} and \ref{prop3}, $\Delta_A$ is a cusp differential operator of order $(2p,2)$. \begin{definition} \label{def7} Let $A$ be a (complex-valued) cusp vector potential. Given a connected component $M_0$ of $M$, we say $A$ is a \emph{trapping} vector potential on $M_0$ if \begin{itemize} \item either the restriction $\varphi_0:=\varphi(0)$ is not constant on $M_0$, \item or $\theta_0:=\theta(0)$ is not closed on $M_0$, \item or the cohomology class $[{\theta_0}_{|M_0}]\in H^1_{\mathrm{dR}}(M_0)$ does not belong to the image of \[2\pi H^1(M_0;\zz)\to H^1(M_0,\cz)\simeq H^1_{\mathrm{dR}}(M_0)\otimes\cz.\] \end{itemize} and \emph{non-trapping on $M_0$} otherwise. We say that $A$ is \emph{trapping} if $A$ is trapping on each connected component of $M$. The vector potential is said to be \emph{non-trapping} if it is not trapping, i.e. there exists a connected component of $M$ on which $A$ is non-trapping on. If $A$ is non-trapping on all connected component of $M$, we say that it is \emph{maximal non-trapping}. \end{definition} We comment briefly the terminology. When $A$ is real, constant near $M$, the multiplicity of the absolutely continuous part of the spectrum of $\Delta_A$ will be given by the number of connected component of $M$ on which $A$ is non-trapping. Hence, taking $A$ maximal non-trapping maximizes the multiplicity of this part of the spectrum. We refer to \cite{gh} for an exposition of cohomology with integer coefficients. The trapping property is determined only by the asymptotic behavior of $A$. More precisely, if $A'$ is also of the form \eqref{aft} with $\varphi(0)=0$ and $\theta(0)=0$ then $A$ is a (non-)trapping vector potential if and only if $A+A'$ is. The term ``trapping" is motivated by dynamical consequences of Theorems \ref{p:pmag} and \ref{t:mourre0} and has nothing to do with the classical trapping condition. This terminology is also supported by the examples given in Section \ref{s:nonstab}. When the first cohomology group $H^1(X;\rz)$ is trivial, we reformulate the trapping condition by some conditions on the magnetic field itself. This is discussed in Section \ref{s:inv}. For a trapping vector potential, $x^{2p}\Delta_A$ is a fully-elliptic cusp operator. In turn, this implies that $\Delta_A$ has empty essential spectrum so from a dynamical point of view, a particle can not diffuse, in other words it is trapped in the interior of $X$. Indeed, given a state $\phi\in L^2(X)$, there exists $\cchi$ (the characteristic function of a compact subset of $X$) such that $1/T\int_0^T \|\cchi e^{it\Delta_A} \phi\|^2 dt$ tends to a positive constant as $T$ goes to infinity. On the other hand, if $A$ is a non-trapping vector potential, then $\Delta_A$ is not Fredholm between the appropriate cusp Sobolev spaces. If the metric is an exact cusp metric and complete, we show that $\Delta_A$ has nonempty essential spectrum also as an unbounded operator in $L^2$, given by $[\kappa(p), \infty)$ by Proposition \ref{p:thema}. We go even further and under some condition of decay of $\varphi$ and $\theta$ at infinity, we show that there is no singular continuous spectrum for the magnetic Laplacian and that the eigenvalues of $\R\setminus \{\kappa(p)\}$ are of finite multiplicity and can accumulate only in $\{\kappa(p)\}$. Therefore given a state $\phi$ which is not an eigenvalue of $\Delta_A$, one obtains that for all $\cchi$, $\cchi e^{it\Delta_A}\phi$ tends to $0$ as $t\to\infty$. When $M$ is connected, the class of non-trapping vector potentials is a group under addition but that of trapping vector potential is not. When $M$ is disconnected, none of these classes is closed under addition. Directly from the definition, we get however: \begin{remark}\label{addtrap} Let $A$ be a maximal non-trapping vector potential and let $A'$ be a smooth vector potential. Then $A'$ is (non-)trapping if and only if $A+A'$ is. \end{remark} Let $B$ be a smooth magnetic field on $\oX$ whose pull-back to $M$ vanishes. Since $B$ is exact, it is also closed, thus it defines a relative de Rham class as in Subsection \ref{rdR}. If this class vanishes, we claim that there exists a vector potential $A$ for $B$ which is maximal non-trapping. Indeed, let $A\in \Lambda^1(\oX,M)$ be any primitive of $B$. $A$ is clearly a cusp form, the singular term $\phi(0)$ vanishes, and the pull-back to each boundary component vanishes, in particular it defines the zero $1$-cohomology class. From Remark \ref{addtrap} we get \begin{corollary}\label{cohcl} Let $B$ be a cusp magnetic field. Let $B'$ be a smooth magnetic field on $\oX$ which vanishes on the boundary and which defines the zero relative cohomology class in $H^2(\oX,M)$. Then $B$ admits (non-)trapping vector potentials if and only if $B+B'$ does. \end{corollary} \section{The trapping case} \subsection{The absence of essential spectrum} In this section, given a smooth cusp $1$-form, we discuss the link between its behavior at infinity and its trapping properties. \begin{theorem}\label{p:pmag} Let $p>0$, $g_p$ a metric on $X$ given by \eqref{pme} near $\partial X$ and $A$ a smooth cusp $1$-form given by \eqref{aft}. Then $\Delta_A$ is of order $(2p,2)$. If $A$ is trapping then $\Delta_A$ is essentially self-adjoint on $\cC_c^\infty(X)$, it has purely discrete spectrum and its domain is $x^{2p}H^2(X,g_p)$. \end{theorem} If $p\leq 1$ then $g_p$ is complete so $\Delta_A$ is essentially self-adjoint \cite{shubin}. This fact remains true for a trapping $A$ in the incomplete case, i.e.\ $p>1$. \proof Using Lemma \ref{prop3}, we get \[\cN(\dA)(\xi)= \begin{bmatrix} d^M+i\theta_0\\ i(\xi+\varphi_0)\end{bmatrix}.\] Suppose that $x^{2p}\Delta_A$ is not fully elliptic, so there exists $\xi\in\rz$ and $0\neq u\in\ker(\cN(x^{2p}\Delta_A)(\xi))$. By elliptic regularity, $u$ is smooth. We replace $M$ by one of its connected components on which $u$ does not vanish identically, so we can suppose that $M$ is connected. Using Lemma \ref{lema6}, by integration by parts with respect to the volume form ${a_0}^{1/2}dh_0$ on $M$ and the metric $h_0$ on $\Lambda^1(M)$, we see that $u\in\ker(\cN(\Delta_A)(\xi))$ implies $u\in\ker(\cN(\dA)(\xi))$. Then \begin{align}\label{xvp} (\xi+\varphi_0)u= 0\mbox{ and }(d^M+i\theta_0)u= 0, \end{align} so $u$ is a global parallel section in the trivial bundle $\ucz$ over $M$, with respect to the connection $d^M+i\theta_0$. This implies \begin{equation*}\label{zudg} 0=(d^M)^2 u=d^M(-iu\theta_0) =-i(d^M u)\wedge\theta_0-iud^M\theta_0=-iud^M\theta_0. \end{equation*} By uniqueness of solutions of ordinary differential equations, $u$ is never $0$, so $d^M\theta_0=0$. Furthermore, from Eq.\ \eqref{xvp}, we see that $\varphi_0$ equals the constant function $-\xi$. It remains to prove the assertion about the cohomology class $[\theta_0]$. Let $\tM$ be the universal cover of $M$. Denote by ${\tilde{u}}$, ${\tilde{\theta}_0}$ the lifts of $u,\theta_0$ to $\tM$. The equation $(d^M+i\theta_0)u=0$ lifts to \begin{equation}\label{ecu} (d^{\tM}+i{\tilde{\theta}_0}){\tilde{u}}=0. \end{equation} The $1$-form ${\tilde{\theta}_0}$ is closed on the simply connected manifold $\tM$, hence it is exact (by the universal coefficients formula, $H^1(\tM,\cz)=H_1(\tM,\cz)=H_1(\tM;\zz)\otimes\cz$, and $H_1(\tM;\zz)$ vanishes as it is the abelianisation of $\pi_1(\tM)$). Let $v\in\cun(\tM)$ be a primitive of ${\tilde{\theta}_0}$, i.e., $d^{\tM}v={\tilde{\theta}_0}$. Then, from \eqref{ecu}, ${\tilde{u}}=Ce^{-iv}$ for some constant $C\neq 0$. The fundamental group $\pi_1(M)$ acts to the right on $\tM$ via deck transformations. The condition that ${\tilde{u}}$ be the lift of $u$ from $M$ is the invariance under the action of $\pi_1(M)$, in other words \[{\tilde{u}}(y)={\tilde{u}}(y[\gamma])\] for all closed loops $\gamma$ in $M$. This is obviously equivalent to \begin{align*} v(y[\gamma])-v(y)\in 2\pi\zz,&&\forall y\in \tilde{M}.\end{align*} Let $\tilde{\gamma}$ be the lift of $\gamma$ starting in $y$. Then \begin{align*} v(y[\gamma])-v(y)=&\int_{\tilde{\gamma}}d^{\tM}v =\int_{\tilde{\gamma}}{\tilde{\theta}_0}=\int_\gamma \theta_0. \end{align*} Thus the solution ${\tilde{u}}$ is $\pi_1(M)$-invariant if and only if the cocycle $\theta_0$ evaluates to an integer multiple of $2\pi$ on each closed loop $\gamma$. These loops span $H_1(M;\zz)$, so $[\theta_0]$ lives in the image of $H^1(M;\zz)$ inside $H^1(M,\cz)=\Hom(H_1(M;\zz),\cz)$. Therefore the solution $u$ must be identically $0$ unless $\varphi_0$ is constant, $\theta_0$ is closed and $[\theta_0]\in 2\pi H^1(M;\zz)$. Conversely, if $\varphi_0$ is constant, $\theta_0$ is closed and $[\theta_0]\in 2\pi H^1(M;\zz)$ then ${\tilde{u}}=e^{-iv}$ as above is $\pi_1(M)$-invariant, so it is the lift of some $u\in\cun(M)$ which belongs to $\ker(\cN(x^{2p}\Delta_A)(\xi))$ for $-\xi$ equal to the constant value of $\varphi_0$. The conclusion of the theorem is now a consequence of general properties of the cusp calculus \cite[Theorem 17]{wlom}. Namely, since $\Delta_A$ is fully elliptic, there exists an inverse in $x^{2p}\Psi_c^{-2}(X)$ (a micro-localized version of $\Diff_c(X)$) modulo compact operators. If $p>0$, this pseudo-inverse is itself compact. The operators in the cusp calculus act by closure on a scale of Sobolev spaces. It follows easily that for $p\geq 0$, a symmetric fully elliptic cusp operator in $x^{2p}\Psi_c^{2}(X)$ is essentially self-adjoint, with domain $x^{2p}H^2_c(X)$. Thus $\Delta_A$ is self adjoint with compact inverse modulo compact operators, which shows that the spectrum is purely discrete. \qed The cusp calculus \cite{meni96c} is a particular instance of Melrose's program of micro-localizing boundary fibration structures. It is a special case of the fibered-cusp calculus \cite{mame99} and can be obtained using the groupoid techniques of \cite{lani}. \subsection{Eigenvalue asymptotics for trapping magnetic Laplacians} If $A$ is a trapping vector potential, the associated magnetic Laplacian has purely discrete spectrum. In this case we can give the first term in the eigenvalue growth law. \begin{theorem}\label{t:thmag} Let $p>0$, $g_p$ a metric on $X$ given by \eqref{pme} near $\partial X$ and $A\in\cun(X,T^*X)$ a complex-valued trapping vector potential in the sense of Definition \ref{def7}. Then the eigenvalue counting function of $\Delta_A$ satisfies \begin{equation}\label{e:thmag} N_{A,p}(\lambda) \approx \begin{cases} C_1\lambda^{n/2}& \text{for $1/n< p<\infty$,}\\ C_2\lambda^{n/2}\log \lambda &\text{for $p=1/n$,}\\ C_3\lambda^{1/2p}&\text{for $0

0$ the form $\theta'_x$ is exact. By the Hodge decomposition theorem, the space of exact forms on $M$ is closed, so the limit $\theta'_0$ is also exact. Now $dw$ is an exact cusp form, in particular it is closed. This implies that $x^2\px\theta'=d^M \varphi'$. Setting $x=0$ we deduce $d^M(\varphi')_{x=0}=0$, or equivalently $\varphi'_{|x=0}$ is constant. Hence the conditions from Theorem \ref{t:thmag} on the vector potential are satisfied simultaneously by $A$ and $A'$. \begin{proof} From Theorem \ref{p:pmag}, the operator $x^{2p}\Delta_A$ is fully elliptic when $A$ is trapping. The result follows directly from \cite[Theorem 17]{wlom}. Let us explain the idea: the complex powers $\Delta_A^{-s}$ belong to the cusp calculus, and are of trace class for sufficiently large real part of $s$. The trace of the complex powers is holomorphic for such $s$ and extends meromorphically to $\cz$ with two families of simple poles, coming from the principal symbol and from the boundary. The leading pole governs eigenvalue asymptotics, by the Delange theorem. In case the leading pole is double (by the superposition of the two families of poles), we get the logarithmic growth law. The explicit computation of the constants (given by \cite[Prop.\ 14 and Lemma 16]{wlom}) is straightforward. \end{proof} \section{The non-trapping case} \subsection{The essential spectrum} We have shown in Theorems \ref{p:pmag} and \ref{t:thmag} that the essential spectrum of the magnetic Laplacian $\Delta_A$ is empty when $A$ is trapping, and we have computed the asymptotics of the eigenvalues. We now consider the case of a non-trapping vector potential $A$, see Definition \ref{def7}. One can guess that in this last case, the essential spectrum is not empty. Let us prove that this is so, in the complete case. We restrict ourselves to conformal perturbations of \emph{exact} cusp metrics. In a fixed product decomposition of $X$ near $M$ we rewrite \eqref{cume} as \begin{equation}\label{cume2} g_0=a \left(\frac{dx}{x^2}+\alpha(x)\right)^2+h(x) \end{equation} where $a\in\cun(\oX)$, $\alpha\in\cun([0,\varepsilon)\times M,\Lambda^1(M))$ and $h\in \cun([0,\varepsilon)\times M, S^2TM)$. By \cite[Lemma 6]{wlom}, the function $a_0:=a(0)$, the metric $h_0:=h(0)$ and the class (modulo exact forms) of the $1$-form $\alpha_0:=\alpha(0)$, defined on $M$, are independent of the chosen product decomposition and of the boundary-defining function $x$ inside the fixed cusp structure. \begin{definition}\label{defex} The metric $g_0$ is called \emph{exact} if $a_0=1$ and $\alpha_0$ is an exact $1$-form. \end{definition} If $\alpha_0=df$ is exact, then by replacing $x$ with the boundary-defining function $x'=x/(1+xf)$ inside the same cusp structure, we can as well assume that $\alpha_0=0$ (see \cite{wlom}). It follows that $g_0$ is quasi-isometric to a cylindrical metric near infinity. \begin{proposition}\label{p:thema} Let $(X,\tilde g_p)$ be a Riemannian manifold with a conformal \emph{exact} cusp metric $\tilde g_p:=(1+\rho)g_p$, where $g_0$ is exact, $g_p=x^{2p}g_0$ and \begin{eqnarray*} \rho\in L^\infty(X; \R), \quad \inf_{x\in X}\rho(x)\pg -1, \quad \rho(x)\rightarrow 0, \mbox{ as } x\rightarrow 0. \end{eqnarray*} Let $A$ be a non-trapping vector potential given by \eqref{aft}. Then \begin{itemize} \item For $0

1$ and $\tilde g_p:=g_p$ is the unperturbed metric given in \eqref{mc}, then every self-adjoint extension of $\Delta_A$ has empty essential spectrum. \end{itemize} \end{proposition} Note that for $p>1$, the unperturbed metric $g_p$ given in \eqref{mc} is essentially of \emph{metric horn} type \cite{lepe}. \proof Using the Weyl theorem, Lemma \ref{l:pertuA} and by changing the gauge, we can suppose without loss of generality that $A$ is of the form \eqref{acon} with $C=0$. We start with the complete case. The essential self-adjointness follows from \cite{shubin}. In the exact case, $g_p$ is quasi-isometric to the metric \eqref{gp'}. Using \cite[Theorem 9.4]{GG} (see Theorem 9.5 for the case of the Laplacian), to compute the essential spectrum we may replace $h(x)$ in \eqref{cume2} by the metric $h_0:=h(0)$ on $M$, extended to a symmetric $2$-tensor constant in $x$ near $M$, and we may set $\rho=0$. By Proposition \ref{p:cut}, computing $\siges(\Delta_A)$ is the same as computing $\siges(\Delta^l_A)$ on $X'$ of Proposition \ref{p:free'}. By the results of Section \ref{s:diago}, this is given on $L^2(\R^+)$ by $\siges(-\Delta+V_p) =[V_p(\infty), \infty)$. Let now $p> 1$. The metric is no longer complete and $(X,g_p)$ is not proper; one can not apply \cite[Theorem 9.4]{GG}. By Lemma \ref{l:indices} and by the Krein formula, all self-adjoint extensions have the same essential spectrum. So it is enough to consider the Friedrichs extension of $\Delta_A$. We now use Propositions \ref{p:free'} and \ref{p:cut}. The operator $D^*D$ is non-negative, so the spectrum of $\Delta_A^0$ is contained in $[\varepsilon^{2-2p}c_0,\infty)$. By Proposition \ref{p:cut}, the essential spectrum does not depend on the choice of $\varepsilon$. Now we remark that $p>1$ implies $c_0\neq 0$. Indeed, the equality would imply that $1/p\in\zz$, which is impossible. Thus by letting $\varepsilon \to 0$ we conclude that the essential spectrum is empty. \qed We have used above a general lemma about compact perturbations of magnetic Laplacians: \begin{lemma}\label{l:pertuA} Let $A$ and $A'$ in $L^\infty(X, T^*X)$ be two magnetic fields on a smooth Riemannian manifold $(X,g)$ (possibly incomplete) with a measurable metric $g$. Suppose that $A-A'$ belongs to $L^\infty_0(X, T^*X)$. Let $\Delta_A={d_A}^*d_A$ and $\Delta_{A'}={d_{A'}}^*d_{A'}$. Then $(\Delta_A+i)^{-1}- (\Delta_{A'} +i)^{-1}$ is compact. \end{lemma} Here, $L^\infty_0(X, T^*X)$ denotes the space of those forms of $L^\infty(X,T^*X)$ which are norm limit of compactly supported forms. Note that this lemma holds without any modification for a $C^1$ manifold equipped with a (RM) structure, see \cite[section 9.3]{GG}. Unlike the result on the stability of the essential spectrum of the (magnetic) Laplacian from \cite[Theorem 9.5]{GG}, we do not ask for the completeness of the manifold. A magnetic perturbation is much less singular than a perturbation of the metric. \proof Note that the form domain of $\Delta_A$ and of $\Delta_A$ is given by $\Gr:=\Dc(d)$, because $A$ and $A'$ are in $L^\infty(X, T^*X)$. We write with a tilde the extension of the magnetic Laplacians to $\Bc(\Gr, \Gr^*)$. We aim to give a rigorous meaning to \begin{equation}\label{33} (\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=(\Delta_A+i)^{-1}(\Delta_{A'}-\Delta_A) (\Delta_{A'}+i)^{-1}. \end{equation} We have $(\Delta_A+i)^{-1*}\Hr\subset\Gr$. This allows one to deduce that ($\Delta_A+i)^{-1}$ extends to a unique continuous operator $\Gr^*\rightarrow\Hr$. We denote it for the moment by $R$. From $R(\Delta_A+i)u=u$ for $u\in \Dc(\Delta_A)$ we get, by density of $\Dc(\Delta_A)$ in $\Gr$ and continuity, $R(\widetilde{\Delta_A}+i)u=u$ for $u\in\Gr$, in particular \begin{equation*} (\Delta_{A'}+i)^{-1}= R(\widetilde{\Delta_A}+i)(\Delta_{A'}+i)^{-1}. \end{equation*} Clearly, \begin{equation*} (\Delta_A+i)^{-1}= (\Delta_A+i)^{-1}(\Delta_{A'}+i)(\Delta_{A'}+i)^{-1}= R(\widetilde{\Delta_{A'}}+i)(\Delta_{A'}+i)^{-1}. \end{equation*} We subtract the last two relations to get \begin{equation*} (\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=R_z(\widetilde{\Delta_{A'}} -\widetilde{\Delta_A})(\Delta_{A'}+i)^{-1} \end{equation*} Since $R$ is uniquely determined as the extension of $(\Delta_A+i)^{-1}$ to a continuous map $\Gr^*\rightarrow\Hr$, one may keep the notation $(\Delta_A+i)^{-1}$ for it. With this convention, the rigorous version of (\ref{33}) that we shall use is: \begin{equation}\label{34} (\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=(\Delta_A+i)^{-1}(\widetilde \Delta_{A'}-\widetilde \Delta_A)(\Delta_{A'}+i)^{-1}. \end{equation} Since $A''=i( A-A')\in L^\infty_0(X, T^*X)$, the Rellich-Kondrakov lemma gives that $A''\wedge$ belongs to $\Kc(\Gr, L^2(X, T^*X))$. Therefore, $\widetilde\Delta_{A'}-\widetilde\Delta_{A} = (A''\wedge)^* d_{A} - d_{A}^*A''\wedge\in \Kc(\Gr,\Gr^*)$. This gives the announced compactness.\qed \subsection{The spectral and scattering theory} In this section, we refine the study of the essential spectrum given in Proposition \ref{p:thema} for non-trapping vector potential. As the essential spectrum arises only in the complete case, we will suppose that $p\leq 1$. We give below our main result in the study of the nature of the essential spectrum and in scattering theory under short-range perturbation. It is a consequence of the Mourre theory \cite{mou} with an improvement for the regularity of the boundary value of the resolvent, see \cite{ggm} and references therein. We treat some conformal perturbation of the metric $\eqref{gp'}$. To our knowledge, this is the weakest hypothesis of perturbation of a metric obtained so far using Mourre theory. Compared to previous approaches, we use a conjugate operator which is local in energy and therefore can be compared directly to a multiplication operator. We believe that this procedure can be implemented to all known Mourre estimates on manifolds to improve the results obtained by perturbation of the metric. We fix $A_{\rm f}$ a non-trapping vector potential of the form \eqref{acon}. By a change of gauge, one can suppose that $A_{\rm f}=\theta_0$ which is constant in a neighborhood of $M$. Let $M_{\rm t}$ (resp.\ $M_{\rm nt}$) be the union of the connected components of $M$ on which $A_{\rm f}$ is trapping (resp.\ non-trapping). Let $L$ be the operator of multiplication by a smooth function $L\geq 1$ which measures the length of a geodesic going to infinity in the directions where $\theta_0$ is non-trapping: \begin{eqnarray}\label{e:L} L(x)=\begin{cases} -\ln(x)& \text{for $p=1$},\\ -\frac{x^{p-1}}{p-1}& \text{for $p< 1$} \end{cases} \end{eqnarray} on $(0, \varepsilon/4)\times M_{\rm nt}$ for small $x$, and $L=1$ on the trapping part $\big((0, \varepsilon /2)\times M_{\rm nt} \big)^c$. Given $s\geq 0$, let $\Lr_{s}$ be the domain of $L^s$ equipped with the graph norm. We set $\Lr_{-s}:=\Lr_{s}^*$ where the adjoint space is defined such that we have $\Lr_{s}\subset L^2(X, \tilde g) \subset \Lr_{s}^*$, for $\tilde g$ defined in the theorem below. Given a subset $I$ of $\R$, let $I_{\pm}$ be the set of complex number $x\pm iy$, where $x\in I$ and $y>0$. The thresholds $\{\kappa(p)\}$ are given in Proposition \ref{p:thema}. For shorthand, perturbations of \emph{short-range} type (resp.\ \emph{trapping} type) are denoted with the subscript ${\rm sr}$ (resp.\ ${\rm t}$); they are supported in $(0, \varepsilon)\times M_{\rm nt}$ (resp.\ in $((0, \varepsilon/2)\times M_{\rm nt})^c$). We stress that the class of ``trapping type'' perturbations is also of short-range nature, in the sense described in Section \ref{s:srlr}, even if no decay is required. This is a rather amusing phenomenon, linked to the fact that no essential spectrum arises from the trapping cusps. The subscript $\rm lr$ denotes \emph{long-range} type perturbations, also with support in $(0, \varepsilon)\times M_{\rm nt}$. We ask such perturbations to be \emph{radial}, i.e.\ independent of the variables in $M$. In other words, a perturbation $W_{\rm lr}$ satisfies $W_{\rm lr}(x,m)= W_{\rm lr}(x,m')$ for all $m,m'\in M$. \begin{theorem}\label{t:mourre0} Fix $\varepsilon >0$. Let $X$ be endowed with the metric $\tilde g = (1+\rho_{\rm sr}+\rho_{\rm lr}+\rho_{\rm t}) g$, where $g=g_p$ is given in \eqref{gp'} for some $0

0$ and a compact operator $K$ such that the inequality \begin{eqnarray}\label{e:T} E_\cJ(T) [T,iS] E_\cJ(T) \geq c E_\cJ(T)+ K \end{eqnarray} holds in the sense of forms in $L^2(X,g)$, for $T=\Delta_{A_{\rm f}}$. Let $W_0$ be $\widetilde\Delta_{A_{\rm f}}$ sent unitarily in $L^2(X,g)$. By Lemma \ref{l:uni}, $W_0\in\cC^{1,1}(S,\Dc(\Delta_{A_{\rm f}}), \Dc(\Delta_{A_{\rm f}})^*)$ is a sum of short and long-range perturbation as described in Section \ref{s:srlr}. In particular, we get $W_0\in\cC^{1}_{\rm u}(S,\Dc(\Delta_{A_{\rm f}}), \Dc(\Delta_{A_{\rm f}})^*)$. By the point (2) of Lemma \ref{l:uni} and \cite[Theorem 7.2.9]{ABG} the inequality \eqref{e:T} holds for $T$ (up to changing $c$ and $K$). We now go into $L^2(X, \tilde g)$ using $U$ defined before Lemma \ref{l:uni}. We write the con\-ju\-ga\-te operator obtained in this way by $\tilde S$. Hence, $\tilde\Delta_{A_{\rm f}}\in\cC^{1,1}(\tilde S, \Dc(\Delta_{\tilde A_{\rm f}}), \Dc(\Delta_{\tilde A_{\rm f}})^*)$ and given $\cJ$ an open interval included in $\sigma_{\rm ess}(H_0)$, there is $c>0$ and a compact operator $K$ such that \begin{eqnarray}\label{e:tildeT} E_\cJ(\tilde T) [\tilde T,i\tilde S] E_\cJ(\tilde T) \geq c E_\cJ(\tilde T)+ K \end{eqnarray} holds in the sense of forms in $L^2(X, \tilde g)$ for $\tilde T=\tilde \Delta_{A_{\rm f}}$. We now add the perturbation given by $A_{\rm sr}, A_{\rm lr}, A_{\rm t}, V_{\rm sr}, V_{\rm lr},$ and $V_{\rm t}$. Note that $H$ has the same domain as $H_0$ and that $(H+i)^{-1}- (H_0+i)^{-1}$ is compact by Rellich-Kondrakov lemma and Lemma \ref{l:pertuA}. By Lemma \ref{l:magne}, we obtain $H\in\cC^{1,1}(S,\Dc(H),\Dc(H)^*)$. As above, the inequality \ref{e:tildeT} is true for $\tilde T=H$. We now deduce the different claims of the theorem. The first comes from \cite[Theorem 7.5.2]{ABG}. The second ones is a consequence of the Virial theorem. For the third point first note that $\Lr_s\subset\Dc(|A|^s)$ for $s\in [0,2]$ by Lemma \ref{l:sa} and use \cite{ggm} for instance (see references therein). Finally, the last point follows from \cite[Theorem 7.6.11]{ABG}. \qed \section{Analysis of the free case}\label{s:free} In this section we concentrate on the unperturbed metric and the simplest non-trapping potential. We localize the computation on the end $X':=(0,\varepsilon) \times M\subset X$. We assume that on $X':=(0,\varepsilon)\times M\subset X$ we have \begin{align} g_p=&x^{2p}\left(\frac{dx^2}{x^4}+h_0\right),\label{gp'}\\ A_{\rm f}=&Cdx/x^2+\theta_0\label{acon} \end{align} where $C$ is a constant, $\theta_0$ is closed and independent of $x$, and the cohomology class $[\theta_0]\in H^1(M)$ is an integer multiple of $2\pi$. By a change of gauge, one may assume that $C=0$. Indeed, it is enough to subtract from $A_{\rm f}$ the exact $1$-form $d(-C/x)$. \subsection{The high and low energy functions decomposition} Set $\dtt:=d^M+i\theta_0\wedge$. We now decompose the $L^2$ space as follows: \begin{equation}\label{e:decomp} L^2(X')=\Hr_{\rm l} \oplus \Hr_{\rm h}, \end{equation} where $\Hr_{\rm l}:=\Kr \otimes \ker(\dtt)$ with $\Kr :=L^2\left((0,\varepsilon), x^{np-2} dx\right)$, and $\Hr_{\rm h}=\Kr\hat{\otimes} \ker(\dtt)^\perp$. We do not emphasize the dependence on $\varepsilon$ for these spaces as the properties we are studying are independent of $\varepsilon$. The subscripts ${\rm l},{\rm h}$ stand for low and high energy, respectively. The importance of the \emph{low energy functions} space $\Hr_{\rm l}$ is underlined by the following: \begin{proposition}\label{p:free'} The magnetic Laplacian $\Delta_{A_{\rm f}}$ stabilizes the decomposition \eqref{e:decomp} of $L^2(X')$. Let $\Delta_{A_{\rm f}}^l$ and $\Delta_{A_{\rm f}}^h$ be the Friedrichs extensions of the restrictions of $\Delta_{A_{\rm f}}$ to smooth compactly supported functions in $\Hr_{\rm l}$, respectively in $\Hr_{\rm h}$. Then $\Delta_{A_{\rm f}}^h$ has compact resolvent, while \begin{eqnarray*} \Delta^l_{A_{\rm f}}=\big(D^*D+c_0^2x^{2-2p}\big)\otimes 1, \end{eqnarray*} where $c_0:=((2-n)p-1)/2$, and $D:=x^{2-p}\px-c_0x^{1-p}$ acts in $\Krke$. \end{proposition} By Proposition \ref{p:cut}, the essential spectrum is pres\-cri\-bed by the space of low energy functions. This kind of decomposition can be found in various places in the literature, see for instance \cite{lott} for applications to finite-volume negatively-curved manifolds. \begin{remark}\label{r:free'} The dimension of the kernel of $\dtt$ equals the number of connected components of $M$ on which $A_f$ non-trapping. Indeed, take such a connected component $M_1$ and let $v$ be a primitive of $\theta_0$ on its universal cover. Then on different sheets of the cover, $v$ changes by $2\pi\zz$ so $e^{iv}$ is a well-defined function on $M_1$ which spans $\ker(\dtt)$. Note that this decomposition is also valid in the trapping case; the low energy functions space is then $0$. Using Proposition \ref{p:cut}, we obtain the emptiness of the essential spectrum for the unperturbed metric and along the way a special case of Theorem \ref{p:pmag} (however we do not recover the full result in this way, because the metric \eqref{pme} \emph{can not} be reached in a perturbative way from the metric \eqref{gp'}). \end{remark} \proof We decompose the space of $1$-forms as the direct sum \eqref{decom}. Recall that $\delta_M$ is the adjoint of $d^M$ with respect to $h_0$. We compute \begin{equation}\label{e:diam}\begin{split} d_{A_{\rm f}}=& \begin{bmatrix} d^M+i\theta_0\wedge\\ x^2\px \end{bmatrix}\\ d_{A_{\rm f}}^*=& x^{-np}\begin{bmatrix}\delta_M-i\theta_0\lrcorner& -x^2\px \end{bmatrix}x^{(n-2)p}\\ \Delta_{A_{\rm f}}=& x^{-2p}(\dtt^*\dtt-(x^2\px)^2-(n-2)px (x^2\px)). \end{split}\end{equation} On the Riemannian manifold $(M,h)$, $\dtt^*\dtt$ is non-negative with discrete spectrum. Since $-(x^2\px)^2-(n-2)px (x^2\px)=(x^2\px)^*(x^2\px)$ is non-negative, one has that $\Delta_{A_{\rm f}}^h\geq \varepsilon^{-2p} \lambda_1$, where $\lambda_1$ is the first non-zero eigenvalue of $\dtt^*\dtt$. By Proposition \ref{p:cut}, the essential spectrum is independent of $\varepsilon$. By letting $\varepsilon \rightarrow 0$ we see that it is empty; thus $\Delta_{A_{\rm f}}^h$ has compact resolvent. The assertion on $\Delta_{A_{\rm f}}^l$ is a straightforward computation.\qed \subsection{Diagonalization of the free magnetic Laplacian}\label{s:diago} In order to analysis the spectral properties of $\Delta_{A_{\rm f}}$ on $(X,g_p)$, where ${A_{\rm f}}$ is given by \eqref{acon} with $C=0$ and $g_p$ by \eqref{gp'}, we go into some ``Euclidean variables''. We concentrate on the complete case, i.e.\ $p\leq 1$. We start with \eqref{e:decomp} and work on $\Kr$. The first unitary transformation is \begin{align*} L^2\left( x^{np-2} dx\right)\to L^2\left( x^{p-2} dx\right)&& \phi\mapsto x^{(n-1)p/2}\phi. \end{align*} Then we proceed with the change of variables $z:=L(x)$, where $L$ is given by \eqref{e:L}. Therefore, $\Kr$ is unitarily sent into $L^2\left((c,\infty), dz\right)$ for a certain $c$. We indicate operators and spaces obtained in the new variable with a subscript $0$. Thanks to this transformation, we are pursuing our analysis on the manifold $X_0=X$ endowed with the Riemannian metric \begin{align}\label{e:gp''} dr^2+h,&&r\to\infty,\end{align} on the end $X_0'=[1/2,\infty)\times M$. The subscript $r$ stands for radial. The magnetic Laplacian is unitarily sent into an elliptic operator of order $2$ denoted by $\Delta_0$. On $\cC^\infty_c(X_0')$, it acts by \begin{eqnarray}\label{e:H_0} \Delta_0:= Q_p\otimes {d_{\tilde\theta}}^*d_{\tilde\theta}+(-\partial^2_r + V_p)\otimes 1, \end{eqnarray} on the completed tensor product $L^2([1/2,\infty), dr)\hat \otimes L^2(M, h)$, where \begin{eqnarray*} V_p(r)=\left\{\begin{array}{ll} ((n-1)/2)^2\\ c_0r^{-2} \end{array}\right. \mbox{ and } Q_p(r)= \left\{\begin{array}{l} e^{2r}\\ ((1-p)r)^{2p/(1-p)} \end{array}\right. \mbox{ for } \left\{\begin{array}{l} p=1\\ p<1. \end{array}\right. \end{eqnarray*} Recall that $c_0$ is defined in Proposition \ref{p:free'} and $d_{\tilde\theta}=d_M+i\theta_0\wedge$. We denote also by $L_0$ the operator corresponding to $L$ in the new variable. It is bounded from below by a positive constant, equals $1$ on the compact part and on the trapping part, and equals $r\mapsto r$ on the non-trapping part. Let $\Hr^s_0:=\Dc((1+\Delta_0)^{s/2})$ for $s>0$. By identifying $\Hr_0$ with $\Hr^*_0$ by the Riesz isomorphism, by duality, we define $\Hr^s_0$ for $s<0$ with $\Hr^{-s}_0$. We need the next well-known fact. \begin{lemma}\label{l:cchi} For every $\gamma\in\cC^\infty_c(X)$, we have that $\gamma:\Hr^s\subset\Hr^s$ for all $s\in\R$ \end{lemma} \proof A computation gives that there is $c$ such that $\|(\Delta_0+i)\gamma\varphi\|\leq c \|(\Delta_0+i)\varphi\|$, for all $\varphi\in\cC^\infty_c(X_0)$. Since $\cC^\infty_c(X_0)$ is a core for $\Delta_0$, we get the result for $s=2$. By induction we get it for $s\in 2\N$. Duality and interpolation give it for $s\in\R$.\qed \subsection{The local conjugate operator}\label{s:conjop} In this section, we construct some conjugate operators in order to establish a Mourre estimate. By mimicking the case of the Laplacian, see \cite{FH}, one may use the following localization of the generator of dilations. Let $\xi\in\cC^\infty([1/2,\infty))$ such that the support of $\xi$ is contained in $[2,\infty)$ and that $\xi(r)=r$ for $r\geq 3$ and let $\tilde \cchi\in\cC^\infty([1/2,\infty))$ with support in $[1,\infty)$, which equals $1$ on $[2,\infty)$. By abuse of notation, we denote $\tilde\cchi\otimes 1\in\cC^\infty(X_0)$ with the same symbol. Let $\cchi:=1-\tilde\cchi$. On $\cC^\infty_c(X_0)$ we set: \begin{eqnarray}\label{e:S0} S_\infty:= -i (\xi \partial_r +\partial_r \xi)\otimes P_0 \tilde\cchi, \end{eqnarray} where $P:= P_{\ker(\dtt)}$. The presence of $P_0$ reinforces the decomposition in low and high energies. \begin{remark}\label{r:sa} By considering a $C_0$-group associated to a vector field on $\R$ like in \cite[Section 4.2]{ABG}, one shows that $-i(\xi \partial_r +\partial_r \xi)$ is essentially self-adjoint on $\cC^\infty_c(\R)$. This $C_0$-group acts trivially away from the support of $\cchi$ and then it is easy to constructed another $C_0$-group $G_0$ which acts like the first one on the $\supp(\cchi)\otimes P_0$ part and trivially on the rest of the space. Now we consider $\overline{S_\infty}$ the generator of the $G_0$. Since $G_0(t)$ leaves invariant $\cC^\infty_c(X_0)$ the Nelson lemma implies that $\overline{S_\infty}$ is the closure of $S_\infty$. In other words, $\overline{S_\infty}$ is essentially self-adjoint on $\cC^\infty_c(X_0)$. This kind of approach has been used in \cite{bouclet} for instance. \end{remark} Therefore, we denote below also by $S_\infty$ the self-adjoint closure of the operator defined by \eqref{e:S0} on $\cC^\infty_c(X_0)$. However this operator is \emph{not} suitable for very singular perturbations like that of the metric considered in this paper. To solve this problem, one should consider a conjugate operator more ``local in energy''. Concerning the Mourre estimate, as it is local in energy for the Laplacian, one needs only a conjugate operator which fits well on this level on energy. Considering singular perturbation theory, the presence of derivatives in \eqref{e:S0} is a serious obstruction, the idea is replacing the conjugate operator with a multiplication operator in the analysis of perturbations, therefore reducing the r\^ole of derivatives within it. The approach has been used for Dirac operators in \cite{GM} to treat very singular perturbation. The case of Schr\"odinger operators is summarized in \cite[Theorem 7.6.8]{ABG}. We set: \begin{eqnarray}\label{e:SR} S_R= \tilde\cchi \big(\Phi_R(-i\partial_r) \xi +\xi \Phi_R(-i\partial_r)\big)\otimes P_0 \tilde\cchi, \end{eqnarray} where $\Phi_R(x):=\Phi(x/R)$ while $\Phi\in\cC^\infty_c(\R)$ and $\Phi(x)=x$ on $[-1,1]$. The operator $\Phi_R(-i\partial_r)$ is defined on the real line by $\Fr^{-1}\Phi_R\Fr$, while $\Fr$ is the unitary Fourier transform. Let us denote by $S_R$ also the closure of the operator. Unlike \eqref{e:S0}, $S_R$ does not stabilize $\cC^\infty_c(X_0)$ because $\Phi_R(-i\partial_r)$ acts like a convolution with a function with non-compact support and therefore destroys the support. This subspace is sent into $\tilde\cchi \Sr(\R)$, the restriction of the Schwartz space. To motivate the subscript $R$, note that $S_R$ tends strongly in the resolvent sense to $S$, as $R$ goes to infinity. We give some properties of $S_R$. The point (2) is essential to be able to replace $S_R$ by $L_0$ in the theory of perturbation. The point (3) is convenient to be able to express a limit absorption principle in terms of $L_0$, which is very explicit. Of course these two points are false for $S_\infty$ and this explains why we can go further in the perturbation theory compared to the standard approach. \begin{lemma}\label{l:sa} We have: \begin{enumerate} \item For all $R\in[1,\infty]$, the closed operator $S_R$ is essentially self-adjoint on $\cC^\infty_c(X)$. \item For $R$ is finite then $L_0^{-2}S_R^2: \cC^\infty_c(X_0) \rightarrow \Dc(\Delta_0)$ extends to a bounded operator in $\Dc(\Delta_0)$. \item For $R$ is finite, $\Dc(L_0^s)\subset \Dc(|S_R|^s)$ for all $s\in[0,2]$. \end{enumerate} \end{lemma} \proof The case $R=\infty$ is discussed in Remark \ref{r:sa}, so assume that $R$ is finite. We compare $S_R$ with $L_0$, defined in Section \ref{s:diago}, which is essentially self-adjoint on $\cC^\infty_c(X)$. Noting that it stabilizes the decomposition \eqref{e:decomp}, we write also by $L_{0,\rm l}$ its restriction to $\Hr_{\rm l}$, this is simply the multiplication by $r$. We compute on $\cC^\infty_c(X_0)$: \begin{eqnarray*} S_R= \tilde\cchi\big(2\Phi_R(-i\partial_r)\xi L_{0,\rm l}^{-1}+ [\xi, \Phi_R(-i\partial_r)] L_{0,\rm l}^{-1} \big)\otimes P_0 \tilde\cchi L_0 \end{eqnarray*} Noting that $\xi L_{0,\rm l}^{-1}$ is bounded and that $\xi'\in L^\infty$ and using Lemma \ref{l:pseudo}, we get $\|S_R \varphi\|\leq a \|L_0 \varphi\|$, for all $\varphi\in\cC^\infty_c(X_0)$. On the other hand, $[S_R, L_0]$ is equal to the bounded operator \begin{eqnarray*} [S_R, L_0]&=& \tilde\cchi\big([\Phi_R(-i\partial_r), L_{0,\rm l}]\xi L_{0,\rm l}^{-1}\big) \otimes P_0 \tilde\cchi L_0\\ &&+ L_0 \tilde\cchi\big( L_{0,\rm l}^{-1} \xi [\Phi_R(-i\partial_r), L_{0,\rm l}] \big) \otimes P_0\tilde\cchi. \end{eqnarray*} This gives $|\langle S_R \varphi, L_0\varphi \rangle -\langle L_0\varphi, S_R\varphi \rangle| \leq b \|L_0^{1/2}\varphi\|^2$, for all $\varphi\in\cC^\infty_c(X_0)$. Finally, one uses \cite[Theorem X.37]{RS2} to conclude that $S_R$ is essentially self-adjoint. On $\cC^\infty_c(X_0)$, we have \begin{eqnarray}\label{e:LS} L^{-2}_0S_R^2= (2\cchi L^{-1}_{0,\rm l}\xi \Phi_R(-i\partial_r)\cchi\otimes P_0 + \cchi L^{-1}_{0,\rm l} [\Phi_R(-i\partial_r), \xi]\cchi\otimes P_0)^2. \end{eqnarray} All these terms are bounded in $L^2(X_0)$ by Lemma \ref{l:pseudo} and by density. We now compute $\Delta_0 L^{-2}_0S_R^2$. By Lemma \ref{l:cchi}, it is enough to show that $\Phi_R(-i\partial_r)$ and $[\Phi_R(-i\partial_r), \xi]$ stabilize the domain $\Delta$ in $L^2(\R)$. The first one commutes with $\Delta$. For the second one, we compute on $\cC^\infty_c(\R)$. Since $[\Phi_R(-i\partial_r), \xi]$ is bounded in $L^2(\R)$, it is enough to show that the commutator $[\Delta,[\Phi_R(-i\partial_r), \xi]]$ is also bounded in $L^2(\R)$. By Bianchi identity we get, that it is equal to $[\Phi_R(-i\partial_r), [\Delta, \xi]] = [\Phi_R(-i\partial_r), 2 \xi'\partial_r + \xi'']= 2\Phi_R(-i\partial_r)\partial_r \xi'-2 \Phi_R(-i\partial_r) \xi'' + 2 \xi'\Phi_R(-i\partial_r) +\xi''$. This is a bounded operator in $L^2(\R)$ and we get point (2). We note that \eqref{e:LS} is bounded in $L^2(X_0)$. Then, since $S_R^2L_0^{-2}$ is also bounded, we get $\|S_R^2\varphi\|^2 \leq c \|L^2_0\varphi\|$ for all $\varphi \in \cC^\infty(X_0)$. Now taking a Cauchy sequence we deduce $\Dc(L^2_0)\subset \Dc(S_R^2)$. An argument of interpolation gives point (3).\qed The aim of this section is the following Mourre estimate. \begin{theorem}\label{t:mourre_0} Let $R\in [1,\infty]$. Then $e^{itS_R} \Hr^2_0\subset \Hr^2_0$ and also $\Delta_0\in \cC^{2}(S_R, \Hr^2_0, \Hr_0)$. Given an interval $\cJ$ inside $\sigma_{\rm ess}(\Delta_0)$, there exist $\varepsilon_R> 0$ and a compact operator $K_R$ such that \begin{eqnarray*} E_\cJ(\Delta_0) [\Delta_0,iS_R] E_\cJ(\Delta_0) \geq (4\inf(\cJ)- \varepsilon_R) E_\cJ(\Delta_0)+ K_R \end{eqnarray*} holds in the sense of forms, and such that $\varepsilon_R$ tends to $0$ as $R$ goes to infinity. \end{theorem} \proof The regularity assumptions follow from Lemmata \ref{l:regu0} and \ref{l:regu}. The left hand side of \eqref{e:mourre_0} is the commutator $[\Delta_0, iS_R]$ in the sense of forms. It extends to a bounded operator in $\Bc(\Hr^2_0, \Hr^{-2}_0)$ since $\cC^\infty_c(X_0)$ is a core for $\Delta_0$. We can then apply the spectral measure and obtain the inequality using Lemma \ref{l:mourre_0}.\qed Compared to the method from \cite[Lemma 2.3]{FH} (for the case $R=\infty$), we have a relatively more direct proof based on Lemma \ref{l:mourre_0}. However this has no real impact on applications of the theory. We now go in a series of Lemmata to prove this theorem. Given a commutator $[A,B]$, we denote its closure by $[A,B]_0$. \begin{lemma}\label{l:commu} For $R=\infty$, the commutators $[\Delta_0,iS_\infty]_0$ and $[[\Delta_0,iS_\infty],iS_\infty]_0$ belong to $\Bc(\Hr^2_0, \Hr_0)$. For $R$ finite, $[\Delta_0,iS_R]_0$ and $[[\Delta_0,iS_\infty],iS_R]_0$ belong to $\Bc(\Hr_0)$. Moreover, if $p=1$, all higher commutators extend to operators in $\Bc(\Hr^2_0, \Hr_0)$ for $R=\infty$ and in $\Bc(\Hr_0)$ for $R$ finite. \end{lemma} \proof Let $\varphi\in\cC^{\infty}_c(X_0')$ such that $\varphi=\varphi_\R\otimes \varphi_M$ where $\varphi_M\in\cC^\infty(M)$ and $\varphi_\R\in\cC^\infty_c([1/2, \infty))$. Note that $P_0\varphi_M\in\cC^\infty(M)$ by the Hodge decomposition, since $d_A^2=0$. Applying the brackets to $\varphi$, by a straightforward computation, we get \begin{eqnarray}\label{e:commu} [\Delta_0,iS_\infty]=-(4\xi'\partial^2_r+4\xi''\partial_r+\xi'''-2V_p')\tilde \cchi\otimes P_0. \end{eqnarray} By linearity and density, we get $\|[\Delta_0,iS_\infty]\varphi\|\leq C\|(\Delta_0+i)\varphi\|$ for all $\varphi\in\cC^\infty_c(X_0')$. Take now $\varphi\in\cC^\infty_c(X_0)$, considering the support of the commutator, we get \begin{eqnarray*} \|[\Delta_0,iS_\infty]\varphi\|&=& \|[\Delta_0,iS_\infty]\widetilde\Xi\varphi\| \\ &\leq& C\|(\Delta_0+i) \varphi\| + \|[\Delta_0, \widetilde\Xi] \varphi\| \leq C'\|(\Delta_0+i) \varphi\|. \end{eqnarray*} where $\widetilde\Xi\in\cC^{\infty}(X)$ with support in $X_0'$ such that $\widetilde\Xi|_{[1,\infty)\times M}=1$.% and $\widetilde\Xi= 1-\Xi$. Therefore, since $\cC^\infty_c(X_0)$ is a core for $\Delta_0$, we conclude $[\Delta_0,iS_\infty]_0\in\Bc(\Hr^2_0, \Hr_0)$. In the same way we compute \begin{eqnarray}\nonumber [[\Delta_0,iS_\infty], iS_\infty]&=& -\big((16(\xi')^2-8\xi\xi'')\partial^2_r + (24 \xi'\xi''-8\xi\xi''')\partial_r \\\label{e:commu2} &&+4 (\xi'')^2+4\xi'''-2 \xi'''' - 4V_p''\big)\tilde\cchi\otimes P_0. \end{eqnarray} and get $[[\Delta_0,iS_\infty],iS_\infty]_0\in\Bc(\Hr^2_0, \Hr_0)$. For $p=1$, the boundedness of higher commutators follows easily by induction ($V_p=0$ in this case). We compute next the commutators of $\Delta_0$ with $S_R$. As above we compute for $\varphi=\varphi_r\otimes\varphi_M$. For brevity, we write $\Phi_R$ instead of $\Phi_R(-i\partial_r)$. As $\Phi_R$ is not a local operator, we first note that this commutator could be taken in the operator sense. Indeed, $\tilde\cchi$ sends $\varphi_r$ to $\cC^\infty_c(\R)$ (note that $[1/2,\infty)$ is injected in a canonical way into $\R$), then $\Phi_R \xi +\xi\Phi_R$ sends to the Schwartz space $\Sr(\R)$ and finally $\tilde\cchi$ sends to $\tilde\cchi\Sr(\R)$ which belongs to the domain of $\Delta_0$. We compute $[\partial_r^2, \tilde\cchi \big(\Phi_R \xi +\xi \Phi_R\big) \tilde\cchi]\otimes P_0$. Against $\varphi_r\otimes \varphi_M$, we have: \begin{eqnarray}\nonumber [\partial_r^2, \tilde\cchi \Phi_R \xi +\xi \Phi_R \tilde\cchi]&=& \Xi [\partial_r^2, \tilde\cchi \Phi_R \xi +\xi \Phi_R \tilde\cchi] \Xi = \Xi [\partial_r^2, \Phi_R r + r\Phi_R ] \Xi +\Psi_\comp \\ \label{e:commu5} &=& 4\Xi \partial_r\Phi_R \Xi +\Psi_\comp = 4\tilde\cchi \partial_r\Phi_R \tilde\cchi +\Psi_\comp, \end{eqnarray} where $\Psi_\comp$ denotes a pseudo-differential operator with compact support such that its support in position is in the interior of $X_0'$. For $p<1$, the potential part $V_p$ arises. We treat its first commutator: \begin{eqnarray} \nonumber [V_p, \tilde\cchi \Phi_R \xi +\xi \Phi_R\tilde\cchi] &=&\tilde\Xi[V_p, 2\Phi_R r -i\Phi'_R]\tilde\Xi +\Psi_\comp \\ \label{e:commu6} &=&\tilde\cchi\big(2[V_p,\Phi_R]\xi-i[V_p, \Phi_R'] \big)\tilde\cchi+\Psi_\comp, \end{eqnarray} where $\Phi_R'=\Phi_R'(-i\partial_r)$. Applying Lemma \ref{l:pseudo}, we get that $[V_p, \Phi_R]\xi$ and $[V_p, \Phi_R']$ are bounded in $L^2(\R)$ also. Therefore, using like above $\tilde{\Xi}$, we get $\|[\Delta_0,iS_R]\varphi\|\leq C\|\varphi\|$ for all $\varphi\in\cC^\infty_c(X_0)$. This implies that $[\Delta_0,iS_R]_0\in\Bc(\Hr_0)$. For higher commutators, the $n$-th commutator with $\tilde\cchi \Phi_R \xi +\xi \Phi_R \tilde\cchi$ is given by $2^n\tilde\cchi \partial_r^n\Phi_R \tilde\cchi +\Psi_\comp$. Note that $\partial_r^n\Phi_R$ is a compactly supported function of $\partial_r$, so the contribution of this term is always bounded. Consider now the second commutator of $V_p$. As above, since we work up to $\tilde\Xi$, $\tilde{\cchi}$ and $\Psi_\comp$, it is enough to show that the next commutator defined on $\Sr(\R)$ extend to bounded operators in $L^2(\R)$. We treat only the most singular part of the second commutator: \begin{eqnarray}\nonumber [[V_p, \Phi_R]r, \Phi'_R r]&=& [[V_p, \Phi_R], \Phi_R r]r + [V_p, \Phi_R][r,\Phi_R]r \\ \nonumber &\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 + \Phi_R [[V_p, \Phi_R], r]r -i [V_p, \Phi_R]\Phi_R'r \\ \nonumber &\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 - \Phi_R [[r, \Phi_R], V_p]r -i [V_p, \Phi_R][\Phi_R',r] -i [V_p, \Phi_R]r \Phi_R' \\ \nonumber &\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 + i \Phi_R [\Phi_R', V_p]r + [V_p, \Phi_R] \Phi_R'' -i [V_p, \Phi_R]r \Phi_R' \end{eqnarray} These terms extend to bounded operators by Lemma \ref{l:pseudo}. \qed The following lemma is the key-stone for the Mourre estimate. \begin{lemma}\label{l:mourre_0} For all $R\in[1,\infty]$, there exists $K\in\Kc(\Hr^2_0, \Hr^{-2}_0)$ and $N_R\in\Bc(\Hr^2_0, \Hr^{-2}_0)$ such that \begin{eqnarray}\nonumber \langle \Delta_0\varphi, iS_R\varphi \rangle+ \langle iS_R \varphi, \Delta_0\varphi \rangle &=& \\ \label{e:mourre_0} && \hspace*{-4cm}4\langle \varphi, (\Delta_0-\inf(\sigma_{\rm ess}(\Delta_0)) \varphi\rangle + \langle \varphi, N_R \varphi\rangle +\langle \varphi, K\varphi\rangle, \end{eqnarray} for all $\varphi\in\cC^\infty_c(X_0)$ and such that $\|N_R\|_{\Bc(\Hr^2_0, \Hr^{-2}_0)}$ tends to $0$ as $R$ goes to infinity. \end{lemma} \proof First note that the essential spectrum of $\Delta_0$ is $[V_p(\infty), \infty)$, by Proposition \ref{p:thema}. We now act in three steps. Let $\Xi$ be like in the proof of Lemma \ref{l:commu} and let $\varphi\in\cC^\infty_c(X_0)$. Since $\tilde\cchi$ and $\Xi$ have disjoint supports, one has for all $R$ that $\langle \varphi, [\Delta_0,iS_R] \varphi\rangle= \langle\widetilde\Xi \varphi, [\Delta_0,iS_R] \widetilde\Xi\varphi\rangle$. For the first step, we start with $R=\infty$. By \eqref{e:commu} and since $\xi'=1$ on $[2,\infty)$, the Rellich-Kondrakov lemma gives \begin{eqnarray}\label{e:Mstep1} \langle \widetilde\Xi \varphi, [\Delta_0,iS_\infty] \widetilde\Xi\varphi\rangle =\langle \widetilde\Xi \varphi, 4(\Delta_0 -V_p(\infty))(1\otimes P_0)\widetilde\Xi \varphi \rangle+ \langle \varphi, K_1 \varphi \rangle \end{eqnarray} for a certain $K_1\in\Kc(\Hr^2_0,\Hr_0^{-2})$. Indeed, $1-\xi', V_p-V_p(\infty), \xi'',\xi''', V_p'$ belong to $\Kc(\Hr^1_0,\Hr_0)$ since they tend to $0$ at infinity. We consider now $R$ finite. We add \eqref{e:commu5} and \eqref{e:commu6}. We have \begin{eqnarray}\nonumber \langle \widetilde\Xi \varphi, [\Delta_0,iS_R] \widetilde\Xi\varphi\rangle &=&\langle \widetilde\Xi \varphi, 4(\Delta_0 -V_p(\infty) - T_R)(1\otimes P_0)\widetilde\Xi \varphi \rangle \\\label{e:Mstep1'} &&+ \langle \varphi, K_2 \varphi \rangle \end{eqnarray} for a certain $K_2=K_2(R)\in\Kc(\Hr^2_0,\Hr_0^{-2})$ and with $T_R= \partial_t(\partial_r-\Phi_R(\partial_r))$. The compactness of $K_2$ follows by noticing that $L_0^{-1}\in \Kc(\Hr^2_0,\Hr_0)$ and that $L_0[V_p, i S_R]_0\in \Bc(\Hr_0,\Hr_0)$, by Lemma \ref{l:pseudo}. We control the size of $T_R$ by showing that $\|\widetilde\Xi T_R (1\otimes P_0)\widetilde\Xi \|_{\Bc(\Hr^2_0, \Hr^{-2}_0)}$ tends to $0$ as $R$ goes to infinity. By Lemma \ref{l:cchi}, one has that $\widetilde\Xi$ stabilizes $\Hr^{\pm 2}_0$, therefore $-\partial_r^2 \widetilde\Xi$ belongs to $\Bc(\Hr^2_0, L^2(\R))$. It remains to note that $(-\partial_r^2+i)^{-2}T_R$ tends to $0$ in norm by functional calculus, as $R$ goes to infinity. The second step is to control the high energy functions part. Consider the Friedrichs extension of $\Delta_0$ on $\Hr_{0,h}:=L^2([1/2, \infty))\otimes P_0^\perp L^2(M)$. We have: \begin{eqnarray}\nonumber \langle \widetilde\Xi \varphi, \Delta_0 P_0^\perp\widetilde\Xi\varphi\rangle&=& \langle (\Delta_0 P_0^\perp+i)^{-1}(\Delta_0 P_0^\perp+i) \widetilde\Xi \varphi, \Delta_0 P_0^\perp\widetilde\Xi\varphi\rangle\\\label{e:Mstep2} &=& \langle \varphi, K_1 \varphi\rangle \end{eqnarray} where $K_1\in\Kc(\Hr_0^2, \Hr_0^{-2})$. Indeed, note first that $\Delta_0 P_0^\perp\tilde\cchi\in\Bc(\Hr_0^2,\Hr_{0,h})$ and that $(\Delta_0 P_0^\perp+i)^{-1} \in\Kc(\Hr_{0,h})$, since $Q_p$ goes to infinity. Therefore the left hand side belongs $\Kc(\Hr_0^2,\Hr_0)$ and the right hand side belongs to $\Bc(\Hr^2_0,\Hr_0)$. The third step is to come back on the whole manifold. It is enough to note that $[\Delta_0,\tilde\cchi]\in\Kc(\Hr^1_0, \Hr_0)$ and to add \eqref{e:Mstep1}, \eqref{e:Mstep1'} with \eqref{e:Mstep2}.\qed We now turn to the regularity assumptions. Lemma \ref{l:C1} will play a central r\^ole. \begin{lemma}\label{l:regu0} For $R\in [1, \infty]$, one has $\Delta_0\in\cC^1(S_R)$ and $e^{itS_R} \Dc(\Delta_0) \subset\Dc(\Delta_0)$. \end{lemma} \proof We start by showing that $\Delta_0\in\cC^1(S_R)$. We check the hypothesis of Lemma \ref{l:C1}. Let $\cchi_n(r):=\cchi(r/n)$ and $\Dr=\cC^\infty_c(X_0)$. Remark that $\supp(\cchi_n')\subset[n,2n]$ and that $\xi \cchi_n^{(k)}$ tends strongly to $0$ on $L^2(\R^+)$, for any $k\geq 1$. By the uniform boundedness principle, this implies that $\sup_n\|\cchi_n\|_{\Dc(H)}$ is finite. Remark \ref{r:sa} and Lemma \ref{l:sa} give that $\Dr$ is a core for $S_R$. Assumption (1) is obvious, assumption (2) holds since $(1-\cchi_n)$ has support in $[2n,\infty)$ and assumption (3) follows from the fact that $H$ is elliptic, so the resolvent of $\Delta_0$ sends $\Dr$ into $\cC^\infty(X_0)$. The point \eqref{e:C1} follows from Lemma \ref{l:mourre_0}. We now show that \eqref{e:comm} is true. Let $\phi\in\cC^\infty(X_0)\cap\Dc(\Delta_0)$. We have $[\Delta_0,\cchi_n]\phi=[\Delta_0,\cchi_n]\tilde\cchi\phi=2 \cchi_n'\partial_r\tilde\cchi\phi + \cchi_n''\tilde\cchi\phi$. We have $iS_\infty[\Delta_0,\cchi_n]\phi= 2\xi\cchi_n'\partial^2_r P_0\tilde \cchi\phi +(4\cchi\xi_n'' +\xi'\cchi_n')\partial_r P_0 \tilde\cchi\phi+ (2\xi\cchi_n'''+\xi'\cchi_n'')P_0 \tilde\cchi\phi$ and, for a finite $R$, we get $iS_R[\Delta_0,\cchi_n]\phi= \tilde\cchi(2\Phi_R(\partial_r)\xi + [\xi, \Phi_R(\partial_r)]) (2\cchi_n'\partial_r P_0\tilde\cchi\phi + \cchi_n'' P_0\tilde\cchi\phi)$. Both terms are tending to $0$ because of the previous remark, Lemma \ref{l:cchi} and the fact that $[\xi, \Phi_R(\partial_r)]$ is bounded by Lemma \ref{l:pseudo}. From that, we can apply the lemma and obtain $H\in\cC^1(S_R)$. By Lemma \ref{l:commu}, we have that $[\Delta_0, iS_R]_0\in\Bc(\Hr_0^2,\Hr_0)$ and \cite[Lemma 2]{GG0} gives that $e^{itA}\Hr_0^2\subset\Hr_0^2$. \qed The invariance of the domain under the group $e^{itS_R}$ implies that $e^{itS_R}\Hr^s_0\subset\Hr^s_0$ for $s\in [-2, 2]$ by duality and interpolation. This allows one to define the class $\cC^k(S_R, \Hr^{s}_0,\Hr^{-s}_0)$ for $s\in [-2,2]$, for instance; we recall that a self-adjoint operator $H$ is in this class if $t\mapsto e^{itS_R}He^{-itS_R}$ is strongly $C^k$ from $\Hr^{s}_0$ to $\Hr^{-s}_0$. \begin{lemma}\label{l:regu} Let $R\in [1, \infty]$. Then $\Delta_0$ belongs to $\cC^2(S_R, \Hr^2_0,\Hr_0)$ for $p\leq 1$. \end{lemma} \proof From Lemma \ref{l:commu}, the commutators with $S_R$ extend to bounded operators from $\Hr^2_0$ to $\Hr_0$. \qed We finally give an estimation of commutator that we have used above. \begin{lemma}\label{l:pseudo} Let $f\in\cC^{0}(\R)$ with polynomial growth, $\Phi_j\in\cC^\infty_c(\R)$ and $g\in\cC^k(\R)$ with bounded derivatives. Let $k\geq 1$. Assume that $\sup_{t\in\R, |s-t|\leq 1}|f(t)g^{(l)}(s)|<\infty$, for all $1\leq l\leq k$. Then the operator $f[\Phi_1(-i\partial_r), [\Phi_2(-i\partial_r)\ldots [\Phi_k(-i\partial_r), g]\ldots]$, defined on $\cC^\infty_c(\R)$, extends also to a bounded operator. \end{lemma} \proof Take $k=1$. We denote with a hat the unital Fourier transform. We get \begin{eqnarray*} (f[\Phi(-i\partial_r), g]\varphi)(t)= \frac{1}{2\pi}\int \widehat{\Phi}(s-t)f(t)(g(s)-g(t))\varphi(s) ds, \end{eqnarray*} for $\varphi\in\cC^\infty_c(\R)$. In order to show that the $L^2$ norm of the left hand side is uniformly bounded by $\|f\|_2$, we separate the integral in $|s-t|$ lower and bigger than one. We start with the first part. Recalling $t^k\widehat\Phi(t)=(-1)^k\widehat{\Phi^{(k)}}(t)$, $\Phi$ is replaced by $\Phi'$ when we divide by $|s-t|$ the term with $g$. Now since $\widehat{\Phi'}\in L^1$, $\sup_{t\in\R, |s-t|\leq 1}|f(t) (g(s)-g(t))/ (s-t)|$ is finite, and the convolution by a $L^1$ function is bounded in $L^2$, we control this part of the integral. We turn to the part $|s-t|\geq 1$. Let $R\in\N$ such that $|f(t)|\leq C(1+ |t|^R)$. We let appear the Fourier transform of $\Phi^{(R+1)}$. Now to conclude, note that $\sup_{s,t\in\R}|(g(s)-g(t))/ (s-t)|$ is finite and that $t\mapsto|f(t)/(s-t)^R|$ is in $L^\infty$ uniformly in $s$. For higher $k$, one repeats the same decomposition and let appear the terms in the $l$-th derivative of $g$ by regrouping terms.\qed \subsection{A short-range and long-range class of perturbations}\label{s:srlr} In the early versions of Mourre theory, one asked for $[[H,S_R], S_R]$ to be $H-$bounded to obtain refined results of the resolvent like the limit absorption principle and the H\"older regularity of the resolvent. In this section, we check the optimal class of regularity $\cC^{1,1}(S_R)$, for $R$ finite. This is a weak version of the two-commutators hypothesis. We refer to \cite{ABG} for definition and properties. This is the optimal class of operators which give a limit absorption principle for $H$ in some optimal Besov spaces associated to the conjugate operator $S_R$. The operator $\Delta_0$ belongs to $\cC^2(S_R)$ by Lemma \ref{l:regu} and therefore also to $\cC^{1,1}(S_R)$. We now consider perturbations of $\Delta_0$ which are also in $\cC^{1,1}(S_R)$. We define two classes. Consider a symmetric differential operator $T: \Dc(\Delta_0)\rightarrow \Dc(\Delta_0)^*$. Take $\theta_{\rm sr}\in\cC_c^\infty((0,\infty))$ not identically $0$; $V$ is said to be \emph{short-range} if \begin{eqnarray}\label{e:sr} \int_1^\infty \Big\|\theta_{\rm sr}\Big(\frac{L_0}{r}\Big) T \Big\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} dr <\infty. \end{eqnarray} and to be \emph{long-range} if \begin{eqnarray}\nonumber \int_1^\infty && \Big\|[T, L_0]\theta_{\rm lr}\Big(\frac{L_0}{r}\Big) \Big\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} + \Big\|\tilde\Xi[T, P_0]L_0\theta_{\rm lr}\Big(\frac{L_0}{r}\Big)\tilde\Xi\Big\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} \\ \label{e:lr} &&+ \Big\|\tilde\Xi[T, \partial_r]L_0\theta_{\rm lr}\Big(\frac{L_0}{r}\Big)\tilde\Xi \Big\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} \frac{dr}{r} <\infty. \end{eqnarray} where $\theta_{\rm lr}$ is the characteristic function of $[1,\infty)$ in $\R$ and where $\widetilde\Xi\in\cC^{\infty}(X)$ with support in $X_0'$ such that $\widetilde\Xi|_{[1,\infty)\times M}=1$. The first condition is evidently satisfied if there is $\varepsilon \pg 0$ such that \begin{eqnarray}\label{e:sr'} \|L^{1+\varepsilon}_0 T\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} \pp \infty \end{eqnarray} and the second one if \begin{eqnarray}\nonumber \|L^{\varepsilon}_0 [T,L_0]\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} &+& \|L^{1+\varepsilon}_0 \widetilde\Xi[T,P_0]\widetilde\Xi\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)} \\ \label{e:lr'} &+& \|L^{1+\varepsilon}_0 \widetilde\Xi[T,\partial_r]\widetilde\Xi\|_{\Bc(\Dc(\Delta_0), \Dc(\Delta_0)^*)}\pp \infty \end{eqnarray} The condition with $P_0$ essentially tells that the non-radial part of $T$ is a short-range perturbation. This is why we will ask the long-range perturbation to be radial. To show that the first class is in $\cC^{1,1}(S_R)$ for $R$ finite, one use \cite[Theorem 7.5.8]{ABG}. The hypotheses are satisfied thanks to Lemmata \ref{l:sa} and \ref{l:L}. Concerning the second class, one shows that $[T, S_R]\in\cC^{0,1}(S_R)$ by using \cite[Proposition 7.5.7]{ABG} (see the proof of \cite[Proposition 7.6.8]{ABG} for instance). For $\Gr=\Dc(\Delta_0^{1/2})$, the short and long-range perturbation of the electric and magnetic perturbations are given by: \begin{lemma}\label{l:magne} Let $V\in L^\infty(X_0)$ and $\tilde A\in L^\infty(X_0, T^*X_0)$. If $\|L^{1+\varepsilon}A\|_\infty\pp \infty$ (respectively $\|L^{1+\varepsilon}V\|_\infty<\infty$) then the perturbation $\big( d_{A_{\rm f}}^* (i\tilde A\wedge) + (i\tilde A\wedge)^* d_{A_{\rm f}})$ (resp.\ $V$) is short-range in $\Bc(\Gr, \Gr^*)$. If $A$ is radial, $\|L^{\varepsilon}A\|_\infty\pp \infty$ and $\|L^{1+\varepsilon}\partial_r A\|_\infty\pp \infty$ (resp.\ $V$ radial, and $\|L^{1+\varepsilon}\partial_r V\|_\infty\pp \infty$) then the same perturbation is long-range in $\Bc(\Gr, \Gr^*)$. \end{lemma} %\proof %Note that $L^{1+ \varepsilon}\tilde A$ and $L^{-\varepsilon } [d_{A_{\rm f}}, % L^{1+ \varepsilon}]$ belong to $\Bc(L^2(X), L^2(X, T^*X))$. \qed We now describe the perturbation of the metric following the two classes. We keep the notation from Theorem \ref{t:mourre0}. We introduce the canonical unitary transformation due to the change of measure. Set $\rho$ to be $\rho_{\rm sr}$, $\rho_{\rm lr}$ or $\rho_{\rm t}$. Let $U$ be the operator of multiplication by $(1+\rho)^{-n/4}$ in $L^2(X, g)$ and $V$ the operator of multiplication by $(1+\rho)^{(2-n)/4}$ in $L^2(X,T^*X, g)$. The operator $U$ is a unitary operator from $ L^2(X, g)$ onto $L^2(X, \tilde g)$ and $V$ a unitary operator from $L^2(X, T^*X, g)$ onto $L^2(X, T^*X, \tilde g)$. \begin{lemma}\label{l:uni} Let $\widetilde\Delta_{A_{\rm f}}$ be the magnetic Laplacian of vector potential $A_{\rm f}$ acting in $L^2(X, \tilde g)$. Let $W_0:= U^{-1}\widetilde\Delta_{A_{\rm f}} U$. Then, \begin{enumerate} \item On $\cC_c^{\infty}(X)$, $W_0$ acts by $U d_{A_{\rm f}}^* V^2 d_{A_{\rm f}} U$. In $L^2(X,g)$, it is essentially self-adjoint and its domain is $\Dc(\Delta_{A_{\rm f}})$. \item One has $(W_0- \Delta_{A_{\rm f}} +i)^{-1}- (\Delta_{A_{\rm f}}+i)^{-1}$ is compact. \item For $\rho_{\rm sr}$ and $\rho_{\rm t}$, $W_0$ is a short-range perturbation of the magnetic Laplacian $\Delta_{A_{\rm f}}$ in the space $\Bc(\Dc(\Delta_{A_{\rm f}}), \Dc(\Delta_{A_{\rm f}})^*)$. \item For $\rho_{\rm lr}$, $W_0$ is a long-range perturbation of $\Delta_{A_{\rm f}}$ in the space $\Bc(\Dc(\Delta_{A_{\rm f}}), \Dc(\Delta_{A_{\rm f}})^*)$. \end{enumerate} \end{lemma} \proof We write $\widetilde\Delta_{A_{\rm f}}$ with the help of the operator $d_{A_{\rm f}}$. Since the manifold is complete, $\widetilde\Delta_{A_{\rm f}}$ is essentially self-adjoint on $\cC^{\infty}_c(X)$. In particular it corresponds to $\tilde d_{A_{\rm f}}^*\tilde d_{A_{\rm f}}$, the Friedrichs extension. This is equal to $U^2 d_{A_{\rm f}}^* V^2 d_{A_{\rm f}}$ in $L^2(X, \tilde g)$. Now remark that $(1+\rho)^\alpha$ stabilizes $\Dc(\Delta_{A_{\rm f}})$, $\Dc(d_{A_{\rm f}})$ and $\Dc(d_{A_{\rm f}}^*)$, for all $\alpha\in\R$ to obtain the first point. We now compare the two operators in $L^2(X, g)$. We compute on $\cC^\infty_c(X)$. \begin{eqnarray}\nonumber D:=W_0 - \Delta_{A_{\rm f}} &=& U^{-1} U^2 d_{A_{\rm f}}^* V^2 d_{A_{\rm f}} U - d_{A_{\rm f}}^* d_{A_{\rm f}} \\ \label{e:D} &=& %&\hspace{-2cm}=&\hspace{-1cm} %U d_{\theta_0}^* V^2d_{\theta_0}(U-1)+ U^{-1}\widetilde\Delta_{A_{\rm f}}(U-1) + U d_{A_{\rm f}}^* (V^2-1) d_{A_{\rm f}}+ (U-1)\Delta_{A_{\rm f}}. \end{eqnarray} We focus on point $(3)$. The two first terms need a justification. We start with the first term. \begin{eqnarray*} L^{1+\varepsilon} U^{-1} \widetilde\Delta_{A_{\rm f}}(U-1)= U^{-1} \big(\widetilde\Delta_{A_{\rm f}} + [L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-1-\varepsilon}\big) L^{1+\varepsilon}(U-1). \end{eqnarray*} Using Lemma \ref{l:L} and the invariance of the domain under $(1+\rho)^\alpha$, we obtain that $(\widetilde\Delta_{A_{\rm f}} + [L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-\varepsilon})^*$ is bounded from $\Dc(\Delta_{A_{\rm f}})$ to $L^2(X, g)$. Again using properties of $(1+\rho)^\alpha$, we get $|\langle \varphi, L^{1+ \varepsilon }U^{-1}\widetilde\Delta_{A_{\rm f}}(U-1) \varphi \rangle |=$ \begin{eqnarray*} |\langle (\widetilde\Delta_{A_{\rm f}} + [L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-1-\varepsilon})^* U^{-1}\varphi, L^{1+\varepsilon }(U-1)\varphi \rangle | \leq c \|(\Delta_{A_{\rm f}}+i) \varphi\|^2, \end{eqnarray*} for all $\varphi\in\cC^\infty_c(X)$. We consider the second term, we have $L^{1+ \varepsilon }\big(U d_{A_{\rm f}}^* (V^2-1) d_{A_{\rm f}}\big)=U \big( d_{A_{\rm f}}^* + [L^{1+ \varepsilon}, d_{A_{\rm f}}^*] L^{-1 - \varepsilon }\big)L^{1+ \varepsilon } (V^2-1) d_{A_{\rm f}}$. Therefore, by Lemma \ref{l:L} and the invariance of $\Dc(d_{A_{\rm f}})$ by $\rho^\alpha$, we obtain $|\langle \varphi, L^{1+ \varepsilon }\big(U d_{A_{\rm f}}^* (V^2-1) d_{A_{\rm f}}\big) \varphi\rangle|=$ \begin{eqnarray*} %\hspace*{-4cm}|\langle \varphi, L^{1+ \varepsilon }\big(U d_{\theta_0}^* %(V^2-1) d_{\theta_0}\big) \varphi\rangle|\hspace*{+4cm}&\hspace*{-8cm}=& \\ |\langle \big( d_{A_{\rm f}} + L^{-1 - \varepsilon } [d_{A_{\rm f}}, L^{1+ \varepsilon}]\big)U \varphi, (V^2-1) d_{A_{\rm f}}\varphi \rangle_{L^2(X,T^*X, g)}| \leq c \|(\Delta_{A_{\rm f}}+i) \varphi\|^2, \end{eqnarray*} for all $\varphi\in\cC^\infty_c(X)$. To finish, use the fact that $\cC^\infty_c(X)$ is a core for $\Delta_{A_{\rm f}}$. The point $(4)$ goes in the same way. We turn to point (2), $W_0$ and $\Delta_{A_{\rm f}}$ have the same domain. We take the proof of Lemma \ref{l:pertuA} replacing $\cG$ with this domain. We then obtain a rigorous version of \eqref{33}. Therefore, it remains to check that $W_0-\Delta_{A_{\rm f}}\in\Kc(\Dc(\Delta_{A_{\rm f}}), \Dc(\Delta_{A_{\rm f}})^*)$. This comes directly using Rellich-Kondrakov Lemma and \eqref{e:D}. \qed Finally, we gather various technicalities concerning the operators $L_0$ and $L$. \begin{lemma}\label{l:L} We have that $dL$ is with support in $(0, \varepsilon)\times M_{\rm nt}$ and $dL= f(x)dx$ where $f:(0, \varepsilon )\rightarrow \R$ such that $f$ is $0$ in a neighborhood of $\varepsilon$ and such that $f(x)= -x^{p-2}$ for $x$ small enough. Moreover: \begin{enumerate} \item The operator $L^{-\varepsilon }d(L^{1+\varepsilon })\wedge$ belongs to $\Bc\big(L^2(X, g), L^2(X,T^*X, g)\big)$ and the commutator $L^{-\varepsilon }[\Delta_{A_{\rm f}}, L^{1+ \varepsilon}]$ with initial domain $\cC^\infty_c(X)$ extends to a bounded operator in $\Bc\big( \Dc(\Delta_{A_{\rm f}}), L^2(X, g)\big)$. \item $e^{itL}\Dc(\Delta_{A_{\rm f}})\subset \Dc(\Delta_{A_{\rm f}})$ and $\|e^{itL}\|_{\Bc(\Dc(\Delta_{A_{\rm f}}))}\leq c (1+t^2)$. \item $L^{-1-\varepsilon }\Dc(\Delta_{A_{\rm f}})\subset \Dc(\Delta_{A_{\rm f}})$. \end{enumerate} \end{lemma} \proof With the diagonalization of Section \ref{s:diago}, the operator $\Delta_{A_{\rm f}}$ is given by \eqref{e:H_0}. The operator $L$ corresponds to the operator $L_0$ of multiplication by $r\otimes 1_{M_{\rm nt}}$ on $(c, \infty)$ in this variable and by $1$ on the rest of the manifold. Hence, points (1) and (3) are easily obtained. Moreover $e^{itL_0}/(1+t^2)$ and its first and second derivative belong to $L^2(X_0)$, uniformly in $t$, from which (2) follows.\qed \section{The non-stability of the essential spectrum and of the situation of limit absorption principle}\label{s:nonstab} In $\rz^n$, with the flat metric, it is well-known that only the behavior of the magnetic field at infinity plays a r\^ole in the computation of the essential spectrum. Moreover, \cite[Theorem 4.1]{KS} states that the non-emptiness of the essential spectrum is preserved by the addition of a bounded magnetic field, even if it can become purely punctual. Concerning a compactly support magnetic field, the essential spectrum remains the same, see \cite{MS}. However, it is well-known that one obtains a long-range effect from it, in other words it acts on particles which have support away from it. In the case of $\rz^n$ with a hole of some kind, this phenomena are of special physical interests and are related to the Aharonov-Bohm effect, see section \ref{s:AB} and references therein. In contrast with the Euclidean setting, Theorem \ref{t:thmag} indicates that in general the essential spectrum may vanish under \emph{compactly supported} perturbations of the magnetic field. In the next sections, we discuss this effect both with and without the hypothesis of gauge invariance, and we investigate the coupling constant effect. \subsection{The case $H^1(X)= 0$.}\label{s:inv} In this section, we assume gauge invariance. We first characterize trapping condition in terms of the magnetic field, see Definition \ref{def7} for the case of a magnetic potential. We recall that if $H^1(X)= 0$, given a magnetic field $B$ the spectral properties of the magnetic Laplacian $\Delta_A$ will not depend on the choice of vector potential $A$ such that $dA=B$. Indeed, given $A, A'$ such that $dA=dA'$, the operators $\Delta_A$ and $\Delta_{A'}$ are unitarily equivalent by a gauge transformation. Therefore, we denote the magnetic Laplacian by $\Delta_B$. Let $p>0$ and let $X$ be a endowed with the metric $g_p$ given by \eqref{pme}. For simplicity, assume that $B$ is a smooth $2$-form on $\oX$ such that its restriction to $X$ is exact. Then there exists $A\in \cun(\oX,T^*\oX)$ such that $B=dA$ (since the cohomology of the de Rham complex on $\oX$ equals the singular cohomology of $\oX$, hence that of $X$). Let \begin{align*}M=\sqcup_{\alpha\in\cA}M_\alpha \end{align*} be the decomposition of the boundary $M$ into its connected components. Set \[\cA_0:=\{\alpha\in\cA; H^1(M_\alpha;\rz)=0\}.\] For some $\cB\subset \cA$ set $M_\cB=\sqcup_{\beta\in\cB}M_\beta$ and consider the long exact cohomology sequence of the pair $(\oX,M_\cB)$ with real coefficients: \[H^1(\oX;\rz)\longrightarrow H^1(M_\cB;\rz)\stackrel{\partial}{\longrightarrow} H^2(\oX,M_\cB;\rz)\stackrel{i}{\longrightarrow} H^2(\oX;\rz)\] Since we assume that $H^1(X;\rz)=0$ it follows that the connecting map $\partial$ is injective. If $B$ vanishes under pull-back to $M_\cB$ then (since it is exact on $X$) it defines a class in $H^2(\oX,M_\cB;\rz)$ which vanishes under the map $i$, so it belongs to the image of the injection $\partial$. We denote by $[B]_\beta$ the component of $[B]$ inside $\partial H^1(M_\beta)\subset H^2(\oX,M_\cB;\rz)$. \begin{definition}\label{magtrap} Let $B$ be a smooth exact $2$-form on $\oX$. Denote by $\cB$ the set of those $\alpha\in\cA$ such that $B$ vanishes identically on $M_\alpha$. The field $B$ is called \emph{trapping} if for each $\beta\in\cB$, the component $[B]_\beta \in\partial H^1(M_\beta)\subset H^2(\oX,M_\cB;\rz)$ is not integral, i.e., it does not live in the image of the map of multiplication by $2\pi$ \[H^2(\oX,M_\cB;\zz)\stackrel{2\pi\cdot}{\longrightarrow}H^2(\oX,M_\cB;\rz),\] and \emph{non-trapping} otherwise. \end{definition} Notice that if $B$ is trapping then $\cB$ must contain the index set $\cA_0$ defined above. In order to apply Theorem \ref{t:mourre0} and Theorem \ref{t:thmag} we use the following lemma: \begin{lemma}\label{l:compsupp} \begin{enumerate} \item \label{compsupp1} Let $A$ be a smooth vector potential on $\oX$ such that $dA=0$ in a neighbourhood of $M=\partial X$. Then there exists a smooth vector potential $A'$, \emph{constant} in a neighborhood of $M$, such that $A=A'$ on $M$ and $d(A-A')=0$. \item \label{compsupp2} Assume that $H^1(\oX,\rz)$ vanishes. Let $B$ be a trapping magnetic field on $\oX$. Then every vector potential for $B$ will be trapping. \item \label{compsupp3} Assume moreover that $H_1(X;\zz)=0$. Let $B$ be non-trapping such that $\imath^*_M B=0$. Then every vector potential for $B$ will be non-trapping. \end{enumerate} \end{lemma} Recall that $\pi_1(X)=0\Longrightarrow H_1(X;\zz)=0 \Longrightarrow H^1_\dR(X)=0$. \begin{proof} \ref{compsupp1}) Let us show that one can choose $A$ to be \emph{constant} near the boundary, in the sense that near $M$ it is the pull-back of a form from $M$ under the projection $\pi:[0,\varepsilon)\times M\to M$, where $\varepsilon$ is small enough. Indeed, $A-\pi^* \imath_M^*A$ is closed on the cylinder $[0,\varepsilon)\times M$ and vanishes when pulled-back to $M$. Now $M$ is a deformation retract of the cylinder, so the map of restriction to $M$ induces an isomorphism in cohomology so the cohomology class $[A-\pi^* \imath_M^*A]\in H^1([0,\varepsilon)\times M)$ must be zero. Let $f\in\cun(\oX,\rz)$ be a primitive of this form for $x\leq \varepsilon/2$, then $A-df$ is the desired constant representative. \ref{compsupp2}) Consider the commutative diagram \begin{equation}\label{cohoz} \begin{CD} H^{1}(M_\cB;\rz)@>{\partial}>>H^{2}(\oX,M_\cB;\rz)@>>>H^2(\oX,\rz)\\ @AA{2\pi\cdot}A@AA{2\pi\cdot}A@AA{2\pi\cdot}A\\ H^{1}(M_\cB;\zz)@>{\partial}>>H^{2}(\oX,M_\cB;\zz)@>>>H^2(\oX,\zz) \end{CD} \end{equation} where the horizontal maps come form the long exact sequence of the pair $(\oX,M)$ and the vertical maps are multiplication by $2\pi$. Let $B$ be trapping and choose a vector potential $A$ for $B$, smooth on $\oX$. We claim that $A$ is trapping. Indeed, $A$ is not closed on the components $M_\alpha$, $\alpha\in\cA\setminus\cB$, while it is closed on $M_\cB$. We note that $\partial[A_{|M_\cB}]=[dA]=[B]$, so $\partial [A_{|M_\beta}]=[B]_\beta$. Assume that for some $\beta\in\cB$, the class $[A_{|M_\beta}]$ were integral. Then using the first square from diagram \eqref{cohoz}, it would follow that $[B]_\beta$ was also integral, contradiction. \ref{compsupp3}) If $B$ is non-trapping, there exists $\beta\in\cB$ and $b\in H^2(\oX,M_\cB,\zz)$ with $[B]_\beta=2\pi b$. The image of $[B]_\beta\in\partial H^1(M_\beta,\rz)$ in $H^2(\oX,\rz)$ is zero, thus $b$ maps to a torsion element in $H^2(\oX,\zz)$. From $H_1(X,\zz)=0$ we see using the universal coefficients theorem \[0\to \Ext(H_1(\oX,\zz),\zz)\to H^2(\oX;\zz)\to \Hom(H_2(\oX,\zz),\zz)\to 0\] that $H^2(\oX;\zz)$ is torsion-free. Thus $b$ comes from some $a\in H^1(M_\cB,\zz)$. By commutativity we have $\partial(2\pi a)=[B]_\beta$. Since $\partial$ is an injection, it follows that every vector potential smooth on $\oX$ for $B$ will define an integral cohomology class on $M_\beta$, thus will be non-trapping. \end{proof} A spectacular example is a compactly supported magnetic field which induces very strong long-range effects. If $H^1(\oX;\zz)=0$ and $B$ be is an exact $2$-form with compact support in $X$, then $B$ is maximal non-trapping (see Definition \ref{def7}) if and only if its class in $H^2(\oX,M;\rz)$ is integral. We will stress more on this aspect in the following section. We can also construct compactly supported magnetic fields for which the consequences of Theorem \ref{t:thmag} hold true. We summarize this fact in the next proposition and give an explicit construction in the proof. \begin{proposition}\label{p:contreex} Let $X$ be the interior of a compact manifold $\oX$ with boundary $M=\partial \oX$, endowed with a conformally cusp metric $g_p$. Assume that $H^1(X)=0$ and $H^1(M_j)\neq 0$ for every connected component $M_j$ of the boundary. Then there exists a non-zero smooth magnetic field $B$ with \emph{compact support} such that the essential spectrum of $\Delta_B$ is empty and such that for $p\geq 1/n$ the growth law of the eigenvalues does not depend on $B$ and is given by \eqref{e:thmag}. Such fields $B$ are generic inside compactly supported magnetic fields. \end{proposition} \proof We construct $A$ like in (\ref{aft}) satisfying the hypotheses of Theorem \ref{t:thmag}. We take $\varphi_0$ to be constant. Let $\psi\in\cun([0, \varepsilon))$ be a cut-off function such that $\psi(x)=0$ for $x\in[3\varepsilon/4,\varepsilon)$ and $\psi(x)=1$ for $x\in[0, \varepsilon/2)$. Since $H^1(M_j)\neq 0$, there exists a closed $1$-form $\beta_j$ on $M$ which is not exact. Up to multiplying $\beta_j$ by a real constant, we can assume that the cohomology class $[\beta_j]\in H^1_{\mathrm{dR}}(M_j)$ does not belong to the image of $2\pi H^1(M_j;\zz)\to H^1(M_j;\rz)\simeq H^1_{\mathrm{dR}}(M_j)$. Let $\beta$ denote the form on $M$ which equals $\beta_j$ on $M_j$. Choose $A$ to be $\psi(x)\beta$ for $\varepsilon>x>0$ and extend it by $0$ to $X$. The magnetic field $B=dA=\psi'(x)dx\wedge\beta$ has compact support in $X$. By Theorem \ref{t:thmag}, $\Delta_A$ has purely discrete spectrum with the Weyl asymptotic law eigenvalues independent of $B$. The relative cohomology class $[B]$ lives in the direct sum $\oplus_j \partial H^1(M_j,\rz)\subset H^2(\oX,M,\rz)$. The field $B$ is non-trapping if at least one of its components in this decomposition lives in the image of $H^1(M_j,\zz)$. Since we assume all $H^1(M_j,\rz)$ to be nonzero, the space of non-trapping magnetic fields is a finite union of subspaces of codimension at least $1$. \qed We now discuss the cohomological hypothesis about $X$ and $M$. This hypothesis can be satisfied in all dimensions greater than or equal to $2$, and different from $3$. In dimension $2$, take $X=\rz^2$ endowed with the metric \eqref{pme}. Consider, for instance, the metric $r^{-2p}(dr^2+d\sigma^2)$ given in polar coordinates. Here $M$ is the circle at infinity and $x=1/r$ for large $r$. Thus $b_1(X)=0$ while $b_1(M)\neq 0$. The product of this manifold with a closed, connected, simply connected manifold $Y$ of dimension $k$ yields an example in dimension $2+k$ with the same properties. Indeed, by the K\"unneth formula, the first cohomology group of $\rz^2\times Y$ vanishes, while $H^1(S^1\times Y)\simeq H^1(S^1)=\zz$. Clearly $k$ cannot be $1$ since the only closed manifold in dimension $1$ is the circle. Thus the dimension $3$ is actually exceptional. For orientable $X$ of dimension $3$, the assumptions $H^1(X)=0$ and $H^1(M)\neq 0$ cannot be simultaneously fulfilled. Indeed, we have the following long exact sequence (valid actually regardless of the dimension of $X$) \begin{equation} \label{les3} H^1(\oX)\stackrel{i_M}{\longrightarrow} H^1(M)\stackrel{\delta}{\longrightarrow} H^2(\oX,M). \end{equation} If $\dim(X)=3$, the spaces $H^1(\oX)$ and $H^2(\oX,M)$ are isomorphic by Poincar\'e duality, hence $H^1(\oX)=0$ implies $H^2(\oX,M)=0$ and so (by exactness) $H^1(M)=0$. It should be possible to build a non-orientable example in dimension $3$ such that one could apply Proposition \ref{p:contreex} but we were not able to construct one. We finally give an example not covered by Lemma \ref{l:compsupp} but such that the conclusion of Theorem \ref{t:thmag} holds. We considered so far magnetic fields on $\overline X$ with vector potential smooth on $\oX$. One may consider also more singular magnetic fields arising from Proposition \ref{p:cuspduB}. \subsubsection{Example} Let $X$ be any conformally cusp manifold, without any cohomological assumptions. Suppose that $B=df\wedge dx/x^2$ where $f$ is a function on $X$ smooth down to the boundary $M$ of $X$. Assume that $f$ is not constant on any connected component of $M$. Then the essential spectrum of the magnetic operator (which is well-defined by $B$ if $H^1(X)=0$) is empty. This follows from the fact that $A:=fdx/x^2$ is trapping. Note that the pull-back to the border of the above magnetic field is zero. \subsection{The coupling constant effect}\label{s:coupling} In flat Euclidean space it is shown in \cite{hn}, under some technical hypotheses, that the spectrum has a limit as the coupling constant tends to infinity. In contrast, in the next example, we exhibit the creation of essential spectrum for periodic values of the coupling constant. We will focus on the properties of $\Delta_{gB}$ for some coupling constant $g\in\R$. In order to be able to exploit the two sides of this work we concentrate here on the metric \eqref{gp'}. We assume that $M$ is \emph{connected}, $H^1(X,\zz)=0$ and $H^1(M)\neq 0$. Given $B$ a magnetic potential with compact support, $g B$ is non-trapping if and only its class in $H^2(\oX,M;\rz)$ is integral. Let $G_B$ be the discrete subgroup of those $g\in\rz$ such that $g B$ is non-trapping. As this subgroup is possibly $\{0\}$, we start with some exact form $B$ which represents a nonzero cohomology class in $H^2(\oX,M;\zz)$; then by exactness of the relative cohomology long sequence, $[B]$ lives in the image of the injection $H^1(M;R)\stackrel{\partial}{\longrightarrow}H^2(\oX,M;\rz)$. With these restrictions, $G_B$ is a non-zero discrete subgroup of $\qz$. Now we apply Theorem \ref{t:thmag} for the trapping case and Theorem \ref{t:mourre0} for the non-trapping case. We obtain that \begin{enumerate} \item For $g\in G_B$, the essential spectrum of $\Delta_{g B}$ is given by $[\kappa(p), \infty)$, where $\kappa(p)$ is defined in Proposition \ref{p:thema}. The spectrum of $\Delta_{g B}$ has no singular continuous part and the eigenvalues of $\R\setminus \{\kappa(p)\}$ are of finite multiplicity and can accumulate only in $\kappa(p)$. \item For $g\notin G_B$, the spectrum $\Delta_{g B}$ is discrete and if $p\geq 1/n$, the asymptotic of the eigenvalues depend neither on $g$ nor on $B$. \end{enumerate} We now describe the long-range effect regarding the coupling constant. Take a state $\phi\in L^2(X)$ such that $\phi$ is not an eigenvalue of the free Laplacian $\Delta_0$ and is located in a energy higher than $\kappa(p)$. Since the Fourier transform of an absolutely continuous measure (comparing to the Lebesgue measure) tends to $0$ at infinity, we obtain, for each $g\in G_{g B}$ and for each $\cchi$ operator of multiplication by the characteristic function of compact support that $\cchi e^{it\Delta_{g B}}\phi\rightarrow 0$ as $t\rightarrow \infty$. If one consider $\cchi$ being 1 above the support of the magnetic field, then after some time the norm of $\phi$ above this zone is arbitrary small. Classically, the particle stops interacting with the magnetic field. Let us denote by $\phi'$ the particle at this moment. Now, if we switch on the interaction with intensity as small as one desires one gets $g\notin G_{g B}$ and then the spectrum of $\Delta_{g B}$ is discrete. Therefore there exists $\cchi'$ operator of multiplication by the characteristic function of compact support such that $1/T \int_{0}^{T} \|\cchi' e^{it\Delta_{g B}}\phi'\|^2 dt$ tends to $0$, as $T\rightarrow \infty$. The particle is caught by the magnetic field even thought they are far from being able to interact classically. In other words, switching on the interaction of the magnetic field with compact support has destroyed the situation of limit absorption principle. This is a strong long-range effect. For the sake of utmost concreteness, take $X=\R^2$ endowed with the metric $r^{-2p}(dr^2+d\theta^2)$ in polar coordinates, for $r$ big enough and $1\geq p>0$. The border $M$ is $S^1$ and $H^2(\overline{X}, M;\R)\simeq \Z$. Then for every closed $2$-form $B$ with compact support and non-zero integral, the group $G_B$ defined above is non-zero. \subsection{The case $H^1(X)\neq 0$, the Aharonov-Bohm effect}\label{s:AB} Gauge invariance does not hold in this case, so one expects some sort of Aharonov-Bohm effect \cite{AB}. Indeed, given two vectors potential arising from the same magnetic field, the associated magnetic Laplacians $\Delta_A$ and $\Delta_{A'}$ might be unitarily in-equivalent. The vector potential acquires therefore a certain physical meaning in this case. In flat $\R^n$ with holes, some long-range effect appears, for instance in the scattering matrix like in \cite{R, RY, RY2, Y}, in an inverse-scattering problem \cite{N,W} or in the semi-classical regime \cite{BR}. See also \cite{helffer} for the influence of the obstacle on the bottom of the spectrum. In all the above cases the essential spectrum remains the same. In light of Proposition \ref{p:contreex}, one can expect a much stronger effect in our context. We now give some examples of magnetic fields with compact support such that there exists a non-trapping vector potential $A$, constant in a neighborhood of $M$, and a trapping vector potential $A'$, such that $dA=dA'=B$. To ease the presentation, we stick to the metric \eqref{gp'}. For $A$, one applies Theorem \ref{t:mourre0} and obtain that the essential spectrum of $\Delta_{A}$ is given by $[\kappa(p), \infty)$, that $\Delta_{A}$ has no singular continuous part and the eigenvalues of $\R\setminus \{\kappa(p)\}$ have finite multiplicity and can accumulate only to $\{\kappa(p)\}$. For $A'$, one applies Theorem \ref{t:thmag} to get the discreteness of the spectrum of $\Delta_{A'}$ and to obtain that the asymptotic of eigenvalues depends neither on $A$ nor on $B$ for $n\geq 1/p$. In the next section, we describe how generic this situation is for hyperbolic manifolds of dimension $2$ and $3$. The easy step is to construct the non-trapping vector potential $A$ constant in a neighborhood of $M$, more precisely we construct $A$ to be maximal non-trapping, see Definition \ref{def7}. Indeed, prescribe a closed $1$-form $\theta$ on $M$ defining integral cohomology $1$-classes on each component of the boundary, and extend it smoothly to $X$, constant in a neighborhood on $M$, like in the proof of Proposition \ref{p:contreex}. The magnetic field $B:=dA$ has then compact support and one may apply Theorem \ref{t:mourre0} for $\Delta_A$. We now construct $A'$ by adding to $A$ a closed form $\alpha$, smooth on $\overline{X}$. Since $\alpha$ is closed, $A'$ and $A$ define the same magnetic field. In light of Remark \ref{addtrap}, $A'$ is trapping if and only if $\alpha$ is. We can then apply Theorem \ref{t:thmag} to $\Delta_{A'}$. It remains to show that closed, trapping $\alpha$ do exist. We start with a concrete example. \begin{example} Consider the manifold $\oX=(S^1)^{n-1}\times [0,1]$ with a metric $g_p$ as in \eqref{gp'} near the two boundary components. Let $\theta_i\in \rz$ be variables on the torus $(S^1)^{n-1}$, so $e^{i\theta_i}\in S^1$. Take the vector potential $A$ to be $0$, it is (maximal) non-trapping. Choose now $\alpha=A'$ to be the closed form $\mu d\theta_1$ for some $\mu\in\rz$. It is constant in a neighborhood of $(S^1)^{n-1}$. The class $[i_M^*(A')]$ is an integer multiple of $2\pi$ if and only if $\mu\in\zz$. In other words, $A'$ is non-trapping if and only if $\mu\in\rz\setminus\zz$. Note that the magnetic field $B$ vanishes. \end{example} In order to show the existence of such $\alpha$ in a more general setting, we assume that the first Betti number of each connected component of the boundary is non-zero. It is enough to find some closed $\alpha$, smooth on $\oX$, which on each boundary component represents a non-zero cohomology class. Then, up to a multiplication by a constant, $\alpha$ will be trapping. When $X$ is orientable and $\dim(X)$ is $2$ or $3$, one proceeds as follows. \begin{proposition}\label{p23} Let $\oX$ be a compact manifold with non-empty boundary $M$. Assume that one of the following hypotheses holds: \begin{enumerate} \item $\dim(X)=2$ and $M$ is disconnected; \item $\dim(X)=2$ and $X$ is non-orientable; \item $\dim(X)=3$, $X$ is orientable and none of the connected components of $M$ are spheres. \end{enumerate} Then there exists a closed smooth form $\alpha\in\Lambda^1(\oX)$, constant in $x$ near the boundary, such that for all connected components $M_j$ of $M$, the class $[\alpha_{|M_j}]\in H^1(M_j;\rz)$ is non-zero. \end{proposition} \proof Any cohomology class on $\oX$ admits a smooth representative $\alpha$ up to the boundary. Moreover since $\alpha$ is closed, one can choose $\alpha$ to be \emph{constant} near the boundary using Lemma \ref{l:compsupp}. Thus, in cohomological terms, the proposition is equivalent to finding a class $[\alpha]\in H^1(\oX)$ whose pull-back to each connected component of $M$ is non-zero, i.e., $H^1(M_j)\ni i_{M_j}[\alpha]\neq 0$. Consider first the case $\dim(X)=2$. If $X$ is non-orientable, $H^2(\oX,M)=0$ so $\delta$ is the zero map. If $X$ is oriented, the boundary components are all oriented circles, so $H^1(M_j)\simeq \rz$; the compactly supported cohomology $H^2(\oX,M)$ is isomorphic to $\rz$ via the integration map, and the boundary map $\delta:H^1(M)\to H^2(\oX,M)$ restricted to $H^1(M_j)$ is just the identity map of $\rz$ under these identifications. Thus the kernel of $\delta$ is made of $\nu$-tuples (where $\nu$ is the number of boundary components) $(a_1,\ldots,a_\nu)$ of real numbers, with the constraint $\sum a_j=0$ in the orientable case. By exactness, this space is also the image of the restriction map $H^1(\oX)\to H^1(M)$. Clearly there exist such tuples with non-zero entries, provided $\nu\geq 2$ in the orientable case. Thus the conclusion follows for $\dim(X)=2$. Assume now that $\dim(X)=3$. Then the maps $i_M$ and $\delta$ from the relative long exact sequence \eqref{les3} are dual to each other under the intersection pairing on $M$, respectively on $H^1(\oX)\times H^2(\oX,M)$: \[\int_M i_M(\alpha)\wedge\beta=\int_X \alpha\wedge\delta \beta.\] These bilinear pairings are non-degenerate by Poincar\'e duality; in particular since the pairing on $H^1(M)$ is skew-symmetric, it defines a symplectic form. It follows easily that the subspace \[L:=i_M(H^1(X))\subset H^1(M)\] is a Lagrangian subspace (i.e., it is a maximal isotropic subspace for the symplectic form). Now the symplectic vector space $H^1(M)$ splits into the direct sum of symplectic vector spaces $H^1(M_j)$, By hypothesis, the genus $g_j$ of the oriented surface $M_j$ is at least $1$ so $H^1(M_j)$ is non-zero for all $j$. It is clear that the projection of $L$ on each $H^1(M_j)$ must be non-zero, otherwise $L$ would not be maximal. Hence, there exists an element of $L=i_M(H^1(X))$ which restrict to non-zero classes in each $H^1(M_j)$, as desired. \qed \begin{remark}\label{r:int} If one is interested in some coupling constant effect, it is interesting to choose the closed form $\alpha$ so that every $[\alpha_{|M_j}]$ are non-zero integral classes. In dimension $2$, this amounts to choosing non-zero integers with zero sum. In dimension $3$, as $L=i_M(H^1(X))=i_M(H^1(X,\zz))\otimes \rz$ is spanned by integer classes, we can find an integer class in $L$ with non-zero projection on all $H^1(M_j)$. For real $g$, the vector potential $g\alpha$ is therefore non-trapping precisely for $g$ in a discrete subgroup $g_0\zz$ for some $g_0\in\qz$. \end{remark} \section{Application to hyperbolic manifolds}\label{hyp} We now examine in more detail how this Aharonov-Bohm effect arises in the context of hyperbolic manifolds of finite volume in dimension $2$ and $3$. These are conformally cusp manifold with $p=1$, with unperturbed metric of the form \eqref{mc} and such that every component $M_j$ of the boundary is a circle when $\dim(X)=2$, respectively a flat torus when $\dim(X)=3$: indeed, outside a compact set, the metric takes the form $g=dt^2+e^{-2t}h$, where $t\in[0,\infty)$, and this is of the form \eqref{mc} after the change of variables $x:=e^{-t}$. We denote by $\oX$ the compactification of $X$ by requiring that $x:=e^{-t}$ be a boundary-defining function for ``infinity''. These manifolds and their boundary components always have non-zero first Betti number. For complete hyperbolic \emph{surfaces} with cusps, every smooth $2$-form $B$ on $\oX$ must be exact because $H^2(\oX;\rz)$ is always zero for a non-closed surface. Call $A$ a smooth primitive of $B$. If $B$ vanishes at $M$ then $A$ is necessarily closed over $M$. In terms of cohomology classes, we have $[B]=\delta_\rz [A_{|M}]$ where $\delta_\rz$ is the connecting morphism from the sequence \eqref{les3} with real coefficients. Notice that $A$ can be chosen to define an integer class on $M$ if and only if $[B]$ is integer. Indeed, $H^2(\oX;\zz)$ is also $0$ for a compact surface with non-empty boundary, so if $[B]$ is integer, it must lie in the image of $\delta_\zz$ for the sequence \eqref{les3} with integer coefficients. Conversely, if $[A]$ is integer, i.e., $[A]=2\pi[A_\zz]$ (see diagram \eqref{cohoz} with $M$ in the place of $M_\cB$, where the horizontal maps are now \emph{surjective}) then $[B]=2\pi\delta_\zz [A_\zz]$ is also integer. We summarize these remarks in the following \begin{corollary} Let $B$ be a smooth $2$-form on the compactification of a complete hyperbolic surface $X$ with cusps, and denote by $[B]\in H^2(\oX,M,\rz)$ its relative cohomology class. \begin{itemize} \item If either $X$ is non-orientable, or $X$ is orientable with at least two cusps, then $B$ admits both trapping and non-trapping vector potentials. \item If $X$ is orientable with precisely one cusp, then $B$ admits only non-trapping vector potentials if $[B]$ is integral, while if $[B]$ is not integral then $B$ admits only trapping vector potentials. \end{itemize} \end{corollary} \begin{proof} First note that $B$ is closed since it is of maximal degree; it is exact since $H^2(\oX)=0$ for every surface with boundary; moreover its pull-back to the $1$-dimensional boundary also vanishes, so $B$ defines a relative de Rham class. By Corollary \ref{cohcl}, the existence of trapping and non-trapping vector potentials depends only on this class. If $X$ is orientable and has precisely $1$ cusp, then the map $\delta:H^1(M)\to H^2(\oX,M)$ is an isomorphism both for real and for integer coefficients. Thus $[B]$ is integer if and only if $[A_{|M}]$ is integer. Since the boundary is connected, $A$ is trapping if and only if the cohomology class of its restriction to the boundary is non-integer. If $X$ is oriented and has at least two cusps, identify $H^2(\oX,M)$ and each $H^1(M_j)$ with $\zz$, so that the boundary map restricted to $H^1(M_j)$ is the identity. We can write $[B]$ first as a sum $\sum\alpha_j$ of non-integer numbers, then also as a sum where at least one term is integer. Let $A$ be a $1$-form on $\oX$ which restricts to closed forms of cohomology class $\alpha_j$ on $M_j=S^1$. Then $B-dA$ represents the $0$ class in $H^2(\oX,M)$, so after adding to $A$ a form vanishing at the boundary, we can assume that $B=dA$. Now when all $\alpha_j$ are non-integers, $A$ is trapping, while in the other case it is non-trapping as claimed. If $X$ is non-orientable, the class $[B]$ vanishes. It is enough to find trapping and non-trapping vector potentials for the zero magnetic field, which is done as in the orientable case. \end{proof} When $\dim(X)=3$, we have: \begin{corollary} Let $X$ be an orientable complete hyperbolic $3$-manifold of finite volume. Then every magnetic field $B$ smooth on the compactification $\oX$ admits trapping vector potentials. Assume that the pull-back of $B$ to the boundary $M$ vanishes. If $X$ has precisely one cusp, then there exists a rational (i.e., containing integer classes) infinite cyclic subgroup $G\subset H^2(\oX,M,\rz)$ so that $B$ admits a non-trapping vector potential if and only if $[B]\in G$. In general, one of the following alternative statements holds: \begin{enumerate} \item Either every magnetic field smooth on $\oX$ and vanishing at $M$ admits a non-trapping vector potential, or \item Generically, magnetic fields smooth on $\oX$ and vanishing at $M$ do not admit non-trapping vector potentials. \end{enumerate} There exists moreover $q\in \zz^*$ such that if $[B]$ is integer, then $qB$ admits non-trapping vector potentials. \end{corollary} \begin{proof} For the existence of trapping vector potentials we use the closed form $\alpha$ from Proposition \ref{p23}. Let $A$ be any vector potential for $B$. It suffices to note that for $u\in\rz$, the form $A+u\alpha$ is another vector potential for $B$, which is trapping on each connected component of $M$ except possibly for some discrete values of $u$. Let $h$ denote the number of cusps of $X$. Both the Lagrangian subspace $L\subset H^1(M)$ and the image space $\partial H^1(M)\subset H^2(\oX,M)$ have dimension $h$. By hypothesis, the cohomology class of $B$ on $\oX$ is $0$ so by exactness of \eqref{cohoz}, the relative cohomology class $[B]$ lives in $\partial H^1(M)$. Assume first that $X$ has precisely one cusp. Let $A$ be a vector potential for $B$ (smooth on $\oX)$. We can change $A$ by adding to it any class in the line $L$ without changing $[B]$. Notice that $H^1(M)=\zz^2$. The line $L$ has an integer generator (given by the image of $H^1(\oX,\zz)\to H^1(M,\zz)$). Without loss of generality, we can assume that $L$ is not the horizontal axis. It follows that the translates of all integer points in $\zz^2$ in directions parallel to $L$ form a discrete subgroup of $\qz$. Thus $B$ admits non-trapping vector potentials if and only if the cohomology class $[B]$ inside the $1$-dimensional image $\partial H^1(M)$ lives inside a discrete subgroup. In particular, if $B$ is irrational (i.e., no positive integer multiple of $B$ is an integral class) then $B$ does not admit non-trapping vector potentials. In the general case, assume first that there exists a boundary component $M_j$ so that $L$ projects surjectively onto $H^1(M_j,\rz)$. Let $A$ be a vector potential for an arbitrary magnetic field $B$ which vanishes at $M$. Let $[A'_j]\in H^1(M_j,\rz) $ be such that $[A]_{H^1(M_j)}+[A'_j]$ is integer. Let $[A']\in L$ be an element whose component in $H^1(M_j)$ is $[A'_j]$. Choose a representative $A'$ and extend it to a smooth $1$-form on $\oX$, constant near the boundary. Then $A+A'$ is a non-trapping vector potential. From the definition of $L$, the form $dA'$ defines the zero class in relative cohomology, so from Corollary \ref{cohcl} we get the assertion on $B$. If the assumption on $L$ is not fulfilled, we claim that \[\hat{L}_j:=L\cap \oplus_{i\neq j}H^1(M_i)\] has dimension $h-1$ for all $j$. Indeed, this dimension cannot be $h$ (since the projection of $L$ on $H^1(M_j)$ is not zero) and it cannot be $h-2$ (since the projection is not surjective). It follows easily that $\hat{L}_j$ is a Lagrangian subspace of $\oplus_{i\neq j}H^1(M_i)$. Let $v$ be a vector in $L\setminus \hat{L}_j$. The component $\hat{v}_j$ of $v$ in $\oplus_{i\neq j}H^1(M_i)$ is clearly orthogonal (with respect to the symplectic form) to $\hat{L}_j$, so by maximality it must belong to $\hat{L}_j$. Thus we may subtract this component to obtain, for each $j$, a non-zero element of $L\cap H^1(M_j)$. These elements may be taken integral since $L$ has integer generators. Since $L=\ker\partial$, it follows that the image of $\partial$ is the direct sum of the images $\partial(H^1(M_j))$. As in the case of only one cusp, we see that $B$ has a non-trapping potential if and only if at least one of the components of $[B]$ in this decomposition belong to a certain cyclic subgroup containing integer classes. Set $q$ to be the least common denominator of the generators of these subgroups for all $j$. If $[B]$ is integer, it follows that every vector potential for $qB$ must be maximal non-trapping. \end{proof} %%%%%%%%%%%%%%% \appendix %%%%%%%%%%%%%%% \section{The $C^1$ condition in the Mourre theory}\label{s:c1} \renewcommand{\theequation}{A.\arabic{equation}} \setcounter{equation}{0} In this appendix, we give a general criteria of its own interest to check the, somehow abstract, hypothesis of regularity $\cC^1$ which is a key notion in the Virial theorem within Mourre's theory, see \cite{ABG} and \cite{GG0}. Let $A$ and $H$ be two self-adjoint operators in a Hilbert space $\Hr$. Suppose that the commutator $[H,iA]$ extends to $\Bc(\Dc(H), \Dc(H)^*)$ and that the following Mourre estimate holds true on a interval $\cI$, i.e.\ there is a constant $c>0$ and a compact operator $K$ such that \begin{eqnarray*} E_\cI(H)[H,iA] E_\cI(H)\geq c E_\cI(H) +K. \end{eqnarray*} By supposing that $H\in\cC^1(A)$ (see lower) or that $e^{itA}\Dc(H)\subset\Dc(H)$, the Virial theorem gives that the point spectrum of $H$ is locally finite on $\cI$. This properties are often omitted to be checked in a manifold context in the Mourre analysis. %, see \cite{FH} for instance. The main reason for this is an omission in the hypotheses given in the Mourre theory in \cite{CFKS}, see \cite{GG0}. Without this hypothesis some dense point spectrum could arise where one would expect that the point spectrum is locally finite. To our knowledge, no result exists to show the $C^1$ regularity in a manifold context. In \cite{bouclet}, one however contours the problem by showing the invariance of the domain condition. Besides giving an abstract criteria to check the $\cC^1(A)$, we will explain how to recover the invariance of the domain from it. Given $z\in\rho(H)$, we denote by $R(r)=(H-z)^{-1}$. For $k\in\N$, we recall that $H\in\cC^k(A)$ if for one $z\notin\sigma(H)$ (then for all $z\notin\sigma(H)$) the map $t\mapsto e^{-itA}R(z)e^{itA}$ is $C^k$ in the strong topology. The $\cC^1(A)$ is a key hypotheses for the Virial theorem for instance see \cite{ABG} and \cite{GG0}. We recall a result following from Lemma 6.2.9 and Theorem 6.2.10 of \cite{ABG}. \begin{theorem}\label{th:abg} Let $A$ and $H$ be two self-adjoint operators in the Hilbert space $\Hr$. The following points are equivalent: \begin{enumerate} \item $H\in\cC^1(A)$. \item for one (then for all) $z\notin \sigma(H)$, there is a finite $c$ such that \begin{eqnarray}\label{e:C1a} |\langle A f, R(z) f\rangle - \langle R(\overline{z}) f, Af\rangle| \leq c \|f\|^2, \quad \forall f\in\Dc(A). \end{eqnarray} \item \begin{enumerate} \item [a.]There is a finite $c$ such that for all $f\in \Dc(A)\cap\Dc(H)$: \begin{eqnarray}\label{e:C1b} |\langle Af, H f\rangle- \langle H f, Af\rangle|\leq c(\|H f\|^2+\|f\|^2), \end{eqnarray} \item [b.] for some (then for all) $z\notin \sigma(H)$, the set $\{f\in\Dc(A) \mid R(z)f\in\Dc(A)$ and $R(\overline{z})f\in\Dc(A) \}$ is a core for $A$. \end{enumerate} \end{enumerate} \end{theorem} Note that, in practice, the condition (3.a) is usually easy and follows with the construction of the conjugate operator. The condition (3.b) could be delicate to check. This is the purpose of the next lemma. This lemma is inspired by \cite{bouclet}.%[Proposition 3.9]. \begin{lemma}\label{l:C1} Let $\Dr$ be a subspace of $\Hr$ such that $\Dr\subset\Dc(H)\cap\Dc(A)$, $\Dr$ is a core for $A$ and $H\Dr\subset\Dr$. Let $(\cchi_n)_{n\in \N}$ be a family of bounded operators such that \begin{enumerate} \item $\cchi_n\Dr\subset \Dr, \quad \slim_{n\rightarrow \infty} \cchi_n =1$ and $\sup_n \|\cchi_n\|_{\Dc(H)}<\infty$. \item $A\cchi_n f \rightarrow Af$, for all $f\in\Dr$, as $n$ go to infinity, \item there is $z\notin \sigma(H)$, such that $\cchi_n R(z) \Dr \subset \Dr \mbox{ and } \cchi_n R(\overline{z}) \Dr \subset\Dr$. \end{enumerate} Suppose also that for all $f\in\Dr$ \begin{eqnarray}\label{e:comm} \lim_{n\rightarrow \infty} A [H, \cchi_n] R(z)f= 0 \mbox{ and } \lim_{n\rightarrow \infty} A [H, \cchi_n] R(\overline{z})f= 0. \end{eqnarray} Finally, suppose that there is a finite $c$ such that \begin{eqnarray}\label{e:C1} \hspace*{1cm} |\langle Af, H f\rangle- \langle H f, Af\rangle|\leq c(\|H f\|^2+\|f\|^2),\quad \forall f\in\Dr. \end{eqnarray} Then one has $H\in\cC^1(A)$. \end{lemma} In (1), $\slim$ stands for strong limit. Note that \eqref{e:comm} is well defined by expanding the commutator $[H,\cchi_n]$ and by using (3) and $H\Dr\subset \Dr$. \proof We take \eqref{e:C1}, by polarization and by applying to $\cchi_n R(\overline{z})f$ and to $\cchi_n R(z)f$, with $f\in\Dr$, we get there is a finite $c$ such that \begin{eqnarray*} |\langle A\cchi_n R(\overline{z})f, H \cchi_n R({z})f\rangle- \langle H \cchi_n R(\overline{z}) f, A\cchi_n R({z}) f\rangle|&\leq& \\ &&\hspace*{-6cm} c\|(H+i)\cchi_n R(\overline{z})f\|\cdot \|(H+i)\cchi_n R(z)f\|, \end{eqnarray*} for all $f\in\Dr$. Condition (1) gives that it is lower that $C\|f\|^2$ for some $C$. We now focus on the left hand side, we commute $H$ and $\cchi_n$. We get: \begin{eqnarray*} &=& | \langle \cchi_n R(\overline{z})f, A\cchi_n f\rangle- \langle A\cchi_n f, \cchi_n R({z}) f\rangle\\ && + \langle \cchi_n R(\overline{z})f, A[H,\cchi_n] R({z})f\rangle- \langle A[H, \cchi_n] R(\overline{z}) f, \cchi_n R({z}) f\rangle|. \end{eqnarray*} Using \eqref{e:comm}, the second line vanishes as $n$ goes to infinity. Taking in account the assumptions (1) and (2), we deduce: \begin{eqnarray*} |\langle R(\overline{z})f, A f\rangle- \langle A f, R({z}) f\rangle | \leq C\|f\|^2, \quad \forall f\in\Dr. \end{eqnarray*} Finally, since $\Dr$ is a core for $A$, we obtain \eqref{e:C1a}. We conclude that $H\in\cC^1(A)$. \qed This lemma is very suitable in a manifold context where it seems hard to check the hypothesis $C^1(A)$ directly. Indeed, these hypotheses are easily satisfied with $H$ being the Laplacian $A$ its conjugate operator, constructed as a localization on the ends of the generator of dilatation like in \cite{FH} for instance. Let $\Dr$ be $\cunc(X)$ and $\cchi_n$ a family of operator of multiplication by smooth cut-off functions with compact support. The fact that $A$ is self-adjoint comes usually by some consideration of $C_0$-group associated to some vector fields and using the Nelson Lemma and the invariance of $\Dr$ under the $C_0$-group give that $\Dr$ is a core for $A$, see Remark \ref{r:sa}. The hypothesis (3) follows then by elliptic regularity. The only point to really check is \eqref{e:comm}. At this point one needs to choose more carefully the family $\xi_n$. We check this lemma in Lemma \ref{l:regu} for the standard conjugate operator and our local version. The invariance of the domain is desirable in order to deal, in a more convenient way, directly with operators and no longer with resolvents. On a manifold the $C_0$-group $e^{itA}$ is not explicit and it could be delicate to deal with the domain of $H$ directly. This condition has been checked in \cite{bouclet} for exponentially large manifolds. However, one may obtain this invariance of domain using \cite{GG0} and a $C^1(A)$ condition. We recall: \begin{lemma} If $H\in \cC^1(A)$ and $[H, iA]:\Dc(H)\rightarrow \Hr$ then $e^{itA}\Dc(H)\subset\Dc(H)$, for all $t\in\R$. \end{lemma} In light of this lemma, one understands better the importance of having some $C^1(A)$ criteria. On one hand, one can easily check the invariance of the domain. On the other hand, if the commutator belongs only to $\Bc(\Dc(H), \Dc(H)^*)$ and not $\Bc(\Dc(H), \Hr)$, one may turn to another version of the Mourre Theory like in \cite{ABG} when $H$ has a spectral gap or like in \cite{GJ, S} in the other case. \section{Finite multiplicity of $L^2$ eigenvalues}\label{fm} In order to classify all the maximal symmetric extension of a given cusp-elliptic operator, see section \ref{s:gene} for definitions. One may defect index, if there are equal and finite, one may deduce that all the maximal symmetric extensions are self-adjoint extensions. By the Krein formulae, one hence obtains that difference of the resolvent of two different extensions is finite rank. This implies by Weyl theorem that the essential spectrum is the same for all self-adjoint extensions moreover, by Birman theorem, the wave operators associated to a pair of two such extensions exist and are complete. On the other hand, note that if these index are not finite, one may have some maximal symmetric extension which are not self-adjoint. \begin{lemma}\label{l:indices} Let $\Delta$, acting on $\cun_c(X)$, be a cusp-elliptic operator in $x^{-2p}\Diff^k(X)$ for some $p,k>0$. Then the dimension of any $L^2$-eigenspace of $\Delta^*$ is finite. \end{lemma} Remember that if the operator $\Delta$ is bounded from below, then the defect indices are the same. This lemma guarantees that they are also finite. This point is not obvious when the manifold is not complete. Of course, this result is based on ideas that can be traced back to \cite{melaps} and which are today quite standard. \proof We start by noticing that $\Delta$ can be regarded as an unbounded operator in a larger $L^2$ space. Namely, let $L^2_{\varepsilon}$ be the completion of $\cunc(X,\Lambda^*X)$ with respect to the volume form $e^{-\frac{2\varepsilon}{x}}dg_p$ for some $\varepsilon>0$. Clearly then $L^2_{\varepsilon}$ contains $L^2$. A distributional solution of $\Delta-\lambda$ in $L^2$ is evidently also a distributional solution of $\Delta-\lambda$ in $L^2_{\varepsilon}$. Thus the conclusion will follow by showing that $\Delta$ has in $L^2_{\varepsilon}$ a unique closed extension with purely discrete spectrum. The strategy for this is by now clear. First we conjugate $\Delta$ through the isometry \begin{align*} L^2_{\varepsilon}\to L^2,&&\phi\mapsto e^{-\frac{\varepsilon}{x}}\phi. \end{align*} We get an unbounded operator $e^{-\frac{\varepsilon}{x}} \Delta e^{\frac{\varepsilon}{x}}$ in $L^2$, which is unitarily equivalent to $\Delta$ (acting in $L^2_{\varepsilon}$). Essentially from the definition, \[\cN(x^{2p}e^{-\frac{\varepsilon}{x}}\Delta e^{\frac{\varepsilon}{x}})(\xi)= \cN(x^{2p}\Delta)(\xi+i\varepsilon).\] The normal operator is a polynomial in $\xi$, in particular it is entire. Then by analytic Fredholm theory \cite[Prop.\ 5.3]{melaps}, the set of complex values of $\xi$ for which $\cN(x^{2p}\Delta)(\xi)$ is not invertible, is discrete. Thus there exists $\varepsilon \pg 0$ such that $\cN(x^{2p}\Delta)(\xi+i\varepsilon)$ is invertible for all $\xi\in\rz$. For such $\varepsilon$ the operator $\Delta$ in $L^2_\varepsilon$ is unitarily equivalent to a fully elliptic cusp operator of order $(k,2p)$ in $L^2$. It is then a general fact about the cusp algebra \cite[Theorem 17]{wlom} that such an operator has a unique closed extension and admits a compact inverse modulo compact operators. In particular, its eigenvalues have finite multiplicity. As noted above, the eigenvalues of $\Delta$ in $L^2$ are included in the corresponding eigenspaces of $\Delta$ in $L^2_\varepsilon$. \qed \section{Stability of the essential spectrum} \label{s:stab} It is well-known that the essential spectrum of an elliptic differential operator on a complete manifold can be computed by cutting out a compact part and studying the Dirichlet extension of the remaining operator on the non-compact part (see e.g., \cite{donelili}). This result is obvious using Zhislin sequences, but the approach from loc.\ cit.\ fails in the non-complete case. For completeness, we give below a proof which has the advantage to hold in a wider context and for a wider class of operator, pseudodifferential operators for instance. We start with a general lemma. We recall that a \emph{Weyl sequence} for a couple $(H,\lambda)$ with $H$ a self-adjoint operator and $\lambda\in\rz$, is a sequence $\varphi_n\in\Dc(H)$ such that $\|\varphi_n\|=1$, $\varphi_n\rightharpoonup 0$ (weakly) and such that $(H-\lambda)\varphi_n\rightarrow 0$, as $n$ goes to infinity. It is well-known that $\lambda\in\sigma_{\rm ess}(H)$ if and only if there is Weyl sequence for $(H,\lambda)$. \begin{lemma}\label{l:cut} Let $H$ be a self-adjoint operator in a Hilbert space $\Hr$. Let $\varphi_n$ be a Weyl sequence for the couple $(H,\lambda)$. Suppose that there is a closed operator $\Phi$ in $\Hr$ such that: \begin{enumerate} \item $\Phi \Dc(H)\subset\Dc(H)$,\item $\Phi(H+i)^{-1}$ is compact, \item $[H,\Phi]$ is a compact operator from $\Dc(H)$ to $\Hr$. \end{enumerate} Then there is $\widetilde{\varphi}_n\in\Dc(H)$ such that $(1-\Phi)\widetilde{\varphi}_n$ is a Weyl sequence for $(H,\lambda)$. \end{lemma} \proof First we note (2) implies that $\Phi\varphi_n$ goes to $0$. Indeed, we have $\Phi\varphi_n=\Phi(H+i)^{-1}\big((H-\lambda)\varphi_n +(i+\lambda) \varphi_n \big)$ and the bracket goes weakly to $0$. Similarly, using (3) we get that $[H,\Phi]\varphi_n\rightarrow 0$. Therefore we obtain $\|(1-\Phi)\varphi_n\|\geq 1/2$ for $n$ large enough. We set $\widetilde\varphi_n=\varphi_n/\|(1-\Phi)\varphi_n\|$. Note that $(1-\Phi)\widetilde\varphi_n\rightharpoonup 0$. Finally, $(H-\lambda)(1-\Phi)\widetilde\varphi_n\rightarrow 0$ since $[H,\Phi]\varphi_n\rightarrow 0$. \qed This shows that the essential spectrum is given by a ``non-compact'' part of the space. We now focus on Friedrichs extension. Given a dense subspace $\Dr$ of a Hilbert space $\Hr$ and a positive symmetric operator on $\Dr$. Let $\Hr_1$ be the completion of $\Dr$ under the norm given by $\Qr(\varphi)^2=\langle H\varphi, \varphi\rangle+\|\varphi\|^2$. The domain of the Friedrichs extension of $H$, is given by $\Dc(H_\Fr)=\{f\in \Hr_1 \mid \Dr\ni g\mapsto \langle Hg,f\rangle+\langle g,f\rangle $ extends to a norm continuous function on $\Hr\}$. For each $f\in\Dc(H_\Fr)$, there is a unique $u_f$ such that $\langle Hg,f\rangle + \langle g,f\rangle = \langle g,u_f\rangle$, by Riesz theorem. The Friedrichs extension of $H$, is given by $H_\Fr f=u_f-f$. It is a self-adjoint extension of $H$, see \cite{RS2}. Let $(X,g)$ be a smooth Riemannian with distance $d$. We fix $K$ a smooth compact sub-manifold of $X$ of same dimension. We endow it with the induced Riemannian metric. We set $X'=X\setminus K$. In the following, we embed $L^2(X')$ in $L^2(X)$. We will need the next definition within the proof. \begin{definition} We say that $\Phi$ is a (smooth) \emph{cut-off function} for $K$ if $\Phi\in\cC_c^\infty(X)$ and $\Phi|_K=1$. We say that it is an $\varepsilon$-cut-off is $\supp(\Phi)\subset B(K,\varepsilon)$. \end{definition} We are now able to give a result of stability of the essential spectrum. \begin{proposition}\label{p:cut} Let $d$ be a differential form of order $1$ on $C^\infty_c(X)\rightarrow C^\infty(X,\Lambda^1)$ with injective symbol away from the $0$ section of the cotangent bundle. We denote by $d_X$ and $d_{X'}$ the closure of $d$ in $L^2(X)$ and $L^2(X')$, respectively. Consider $\Delta_X:=d_X^*d_X$ and $\Delta_{X'}=d_{X'}^*d_{X'}$, the Friedrichs extensions of the operator $d^*d$, acting on $\cC_c^\infty(X)$ and $\cC_c^\infty(X')$, respectively. One has $\sigma_{\rm{ess}}(H_X)= \sigma_{\rm{ess}}(H_{X'})$. \end{proposition} \proof Let $f\in L^2(B(K, \varepsilon)^c)$ and let $\Phi$ be a $\varepsilon$-cut-off for $K$. We first show that $f\in\Dc(d_X)$ if and only if $f\in\Dc(d_{X'})$. Suppose that $f\in\Dc(d_X)$, then for all $\eta>0$, there is $\varphi\in \cC_c^\infty(X)$ such that $\|f-\varphi_n\|+ \|df-d\varphi_n\|<\eta$. Because of the support of $f$, one obtain that $\|\Phi\varphi\|<\eta\|\Phi\|_\infty$ and that $\|[d,\Phi]\varphi\|<\eta\|[d,\Phi]\|_\infty$. Therefore $(1-\Phi)\varphi_n\in \cC_c^\infty(X')$ and is Cauchy in $\Dc(d_{X'})$, endow with the graph norm. By uniqueness of the limit, one obtains that $f\in\Dc(d_{X'})$ and that $d_X f=d_{X'}f$. The opposite implication is obvious. Using again the $\varepsilon$-cut-off, one shows that $g\in\Dc(d^*_X)$ if and only if $g\in\Dc(d^*_{X'})$ and that $d^*_{X}g= d^*_{X'}g$ for $g\in L^2(\Lambda^1 (B(K, \varepsilon)^c)$. Finally, we obtain that $f\in\Dc(H_X)$ if and only if $f\in\Dc(H_{X'})$ and that $H_X f=H_{X'} f$, for $f\in L^2(B(K, \varepsilon)^c)$. From the definition of the Friedrichs extension and the injectivity of the symbol of $d$, the domain of $\Delta_X$, $\Delta_{X'}$ is contained in $H^1_0(X)\cap H^2_\loc(X)$, respectively in $H^1_0(X')\cap H^2_\loc(X')$. By taking the same $\Phi$ as above and using the Rellich-Kondrakov lemma, the hypotheses of Lemma \ref{l:cut} are satisfied. 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