Content-Type: multipart/mixed; boundary="-------------0005090504839" This is a multi-part message in MIME format. ---------------0005090504839 Content-Type: text/plain; name="00-215.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-215.keywords" Krein formula Self-adjoint extensions Singular Perturbations ---------------0005090504839 Content-Type: application/x-tex; name="pert.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pert.tex" \documentstyle{amsppt} \TagsAsMath \magnification=\magstep1 \hsize=15truecm \vsize=23truecm \def\d{\frac{d{\ }}{dz}\,} \def\H{\Cal H} \def\X{\Cal X} \def\Y{\Cal Y} \def\Z{\Cal Z} \def\Q{\Cal Q} \def\R{\Cal R} \def\G{\Cal G} \def\K{\Cal K} \def\E{\Cal E} \def\F{\Cal F} \def\RE{\Bbb R} \def\C{{\Bbb C}} \def\GB{\breve G(z)} \def\LD{L^2(\RE^3)} \def\sob#1#2{H^{#1}(\RE^{#2})} \def\sobc#1{H^{#1}(C)} \def\ld#1{L^2(\RE^#1)} \def\HD{H^{2}(\RE^3)} \def\p{\par\noindent} \def\v{\vskip 8pt\p} \def\vp{\varphi} \def\qed{\hfill$\square$\p} \def\proof{\v{\it Proof.} } \def\re{\text{\rm Re}} \def\uno{1@!@!@!@!@!@!\!\text{\rm I}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\def\uno{1@!@!@!@!@!@!\!\text{\rm I}} %\def\NA{{\Bbb N}} %\def\LS{{L^2_{*}} (\Re^3)} %\def\LT{{L^2_{3}} (\RE^3)} %\def\HS{H^s(\RE^3)} %\def\sob#1{H^{#1}(\RE)} %\def\LT{L^2(\RE^3)} %\def\LQ{L^2(\RE^4)} %\def\E{{\Cal E}} %\def\S{{\Cal S}} %\def\Q{{\Cal Q}} %\def\SU{\S(\RE)} %\def\ST{\S(\RE^3)} %\def\lr{\lim_{r\downarrow 0}} %\def\H{H^1(\RE^3)} %\def\HDS{H^{2}_{*}(\RE^3)} %\def\HUT{H^{1}_{3}(\RE^3)} %\def\HDT{H^{2}_{3}(\RE^3)} %\def\CH{{\Cal H}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf A KREIN--LIKE FORMULA FOR SINGULAR PERTURBATIONS} \centerline{\bf OF SELF--ADJOINT OPERATORS AND APPLICATIONS} \vskip 15pt \centerline{\sl by} \vskip 15pt \centerline{\sl Andrea Posilicano} \vskip 5pt \centerline{ Dipartimento di Scienze, Universit\`a dell'Insubria} \centerline{ Via Lucini 3, I-22100 Como, Italy} \centerline{ e-mail: \tt posilicano\@mat.unimi.it} \vskip 50pt \p {\bf Abstract.} {\sl Given a self-adjoint operator $A:D(A)\subset\H\to\H$ and a continuous linear operator $\tau:D(A)\to\X$ with Range$\,(\tau')\cap\H' =\left\{0\right\}$, $\X$ a Banach space, we explicitly construct a family $A^\tau_\Theta$ of self-adjoint operators such that any $A^\tau_\Theta$ coincides with the original $A$ on the kernel of $\tau$. Such a family is obtained by giving a Kre\u\i n-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$. The parameter $\Theta$ belongs to the space of symmetric operators from $\X'$ to $\X$. In the case $\X=\C$ one recovers the ``$\,\H_{-2}$--construction'' of Kiselev and Simon [KS] and so, to some extent, our results can be considered as an extension of it to the infinite rank case. Various applications to singular perturbations of non necessarily elliptic pseudo-differential operators are given, thus unifying and extending previously known results.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 50pt \p {\bf 1. INTRODUCTION}\vskip 10pt\p Let $A:D(A)\to\H$ be a self-adjoint operator on the Hilbert space $\H$ and suppose that there exists a linear dense set $N\subsetneq D(A)$ which is closed with respect to the graph norm on $D(A)$. If we denote by $A_N$ the restriction of $A$ to $N$, then $A_N$ is a closed, densely defined, symmetric operator. Since $N\subsetneq D(A)$, $A$ is a non-trivial extension of $A_N$ and so, by the von Neumann theory on self-adjoint extensions of closed symmetric operators (see [N], [DS, \S XII.4], [RS, \S X.1]), we know that the deficiency indices $n_\pm$, defined as the dimensions of $K_\pm:=$Kernel($A_N^*\pm i$), are equal and strictly positive. The family of self-adjoint extensions of $A_N$ is then parametrized by the unitary maps from $K_+$ onto $K_-$. When $A$ is strictly positive, a deeper and more explicit construction of the (positive if dim $(K)=+\infty$, $K:=$Kernel($A_N^*$)) self-adjoint extensions of $A_N$ is given by the Birman--Kre\u\i n--Vishik theory (see [K3], [V], [B], [AS]). In this case the family of (positive) extensions is parametrized by the (positive) quadratic forms on $K$. \par Any self-adjoint extension $\tilde A_N\not= A$ can then be interpreted as a singular perturbation of $A$ since the two operators differ only on $N^c$, the set $N^c$ being ``thin'' since its complement is a linear dense subset of $\H$.\par In the case $n_{\pm}=1$, Kre\u\i n obtained, in 1943 (see [K1]), a quite explicit formula relating the resolvents of any two self-adjoint extensions of a given symmetric operator. Such a formula was then extended, by Kre\u\i n himself in 1946 (see [K2]), to the case $n_\pm=m<+ \infty$. In our setting it states the following: $$ \forall\,z\in\rho(A)\cap\rho(\tilde A_N),\qquad (-\tilde A_N+z)^{-1}=(-A+z)^{-1} +\sum_{j,k=1}^m\Gamma(z)^{-1}_{jk}\,\varphi_j(z)\otimes\varphi_k(z^*)\, , $$ where $$ \forall\,z\in\rho(A),\qquad \varphi_k(z):=\varphi_k-(i-z)(-A+z)^{-1}\varphi_k\ , $$ $\left\{\varphi_k\right\}_1^m$ being the set of linear independent solutions of $$ A^*_N\varphi=i\,\varphi\,,\qquad\varphi\in D(A^*_N)\ , $$ and where the invertible matrix $\Gamma(z)$ satisfies ($\langle\cdot,\cdot\rangle$ denoting the scalar product on $\H$) $$ \forall\,w,z\in\rho(A)\cap\rho(\tilde A_N),\qquad \Gamma(z)_{jk}-\Gamma(w)_{jk} =(z-w)\langle\varphi_j(z^*),\varphi_k(z)\rangle\ . $$ By such a formula, since $N$ is dense, one can then readily define $\tilde A_N$ as $$ D(\tilde A_N):=\{\phi\in\H\ :\ \phi=\phi_z+\sum_{j,k=1}^{m}\Gamma(z)_{jk}^{-1}\langle\varphi_k(z), \phi_z\rangle\,\varphi_j(z)\,,\ \phi_z\in D(A)\,\} $$ $$ (-\tilde A_N+z)\phi:=(-A+z)\phi_z\ . $$ Kre\u\i n's original papers were written in russian, but his results were popularized in some excellent monographs (see e.g. [AG, chap. VII]). Instead, the analogous formula for the case $n_\pm=+\infty$, which was obtained by Saakjan in 1965 (see [S]), is much less known, since the work is not available in english (see however [GMT] and references therein). Due probably to this fact, the Kre\u\i n formula for $n_\pm=+\infty$ (similar considerations also apply to the Birman--Kre\u\i n--Vishik theory) was rarely used in concrete applications: we are mainly referring to the much studied case of singular perturbations of the Laplacian supported by null sets (see e.g. [AGHH], [AFHKL], [Br] and references therein). Indeed in situations of this kind other approaches are used: extensions are mainly obtained either as resolvent limits of less singular perturbations or by other constructions often resembling variations of either the Kre\u\i n formula or the Birman--Kre\u\i n--Vishik theory. Usually such approaches rely on the elliptic nature of the Laplacian and are not applicable to the study of singular perturbations of hyperbolic operators (this was the original motivation of our work).\par Here we show how, when the (not necessarily dense) set $N$ is the kernel of a continuous linear map $\tau: D(A)\to \X$ such that Range$(\tau')\cap\H'= \left\{0\right\}$, $\X$ a Banach space, one can prove, by almost straightforward arguments, a Kre\u\i n--like formula for self-adjoint extensions of $A_N$, where the role of $K_\pm$ is played by the dual pair $(\X,\X')$ (our construction could be given for $\X$ a locally convex space, but we will not strive here for the maximum of generality). \par The formula given here turns out to be relatively simple: for example we do not need to compute $A_N^*$; in more detail (see theorem 1) one obtains, under a hypothesis which we prove to be satisfied under relatively weak conditions (see proposition 1), and denoting by $A^\tau_\Theta$ the family of self-adjoint extensions of $A_N$, $$ (-A^\tau_\Theta+z)^{-1}=(-A+z)^{-1}+G(z)\cdot(\Theta+\Gamma(z))^{-1}\cdot\breve G(z)\ , $$ where $$ \breve G(z):=\tau\cdot (-A+z)^{-1}\,,\qquad G(z):=C_\H^{-1}\cdot\breve G(z^*)' $$ ($C_\H$ being the canonical isomorphism of $\H$ onto $\H'$) and the conjugate linear operator $\Gamma(z):D\subseteq\X'\to \X$ satisfies the equation $$ \forall\, \ell\in D,\qquad \d \Gamma(z)\ell=\breve G(z)\cdot G(z)\ell $$ which (see lemma 2) we show to have an explicit (in terms of $\tau$ itself) bounded operator solution. Such a solution plays a fundametal role in finding (see lemmata 3 and 4) other nicer (even if unbounded) solutions which we then use in (some of) the examples.\par In \S 3, after showing (example 1) how our construction, in the case $\X=\C$, reproduces the ``$\,\H_{-2}$--construction'' given in [KS] and how, in the case $A$ is strictly positive, it gives a variation on the Birman--Kre\u\i n--Vishik theory which comprises the results in [KKO] (example 2), we use the above Kre\u\i n--like formula to study singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results. More precisely we give the following examples:\p \smallskip \item{--} Finitely many point interaction in three dimensions (example 3); \smallskip \item{--} Infinitely many point interaction in three dimensions (example 4); \smallskip \item{--} Singular perturbations of the Laplacian in three and four dimensions supported by regular curves (example 5); \smallskip \item{--} Singular perturbations, supported by null sets with Hausdorff co-dimension less than $2s$, of translation invariant pseudo-differential operators with domain $H^{s}(\RE^n)$ (example 6); \smallskip \item{--} Singular perturbations of the d'Alembertian in four dimensions supported by time-like straight lines (example 7. In order to limit the lenght of the paper we content ourselves with discussing here only the case of a straight line. A complete study of the case of a generic time-like curve will be the subject of a separate paper. We belive that the detailed study of such a kind of operators will lead to a rigorous framework for the classical and quantum electrodynamics of point particles in the spirit of the results obtained, for the linearized (or dipole) case, in [NP1--3] and [BNP]); \smallskip \item{--} Singular perturbations, supported by null sets, of translation invariant pseudo-differential operators with domain the H\"ormander spaces $H_\vp(\RE^n)$ (example 8). %\vfill\eject \vskip 20pt\p \centerline{\bf Definitions and notations} \vskip 10pt\p \item{$\bullet$} Given a Banach space $\X$ we denote by $\X'$ its strong dual; %\smallskip %\item{$\bullet$} The sequence $\left\{e_n\right\}_1^\infty\subset\X$ %is said to be a Schauder base in the Banach space $\X$ if there %exists a corresponding sequence (the dual one) %$\left\{\ell_n\right\}_1^\infty\subset\X'$ such that %$\ell_i(e_j)=\delta_{ij}$ and $x=\sum_{n=1}^{\infty}\ell_n(x)\,e_n$ for any $x\in\X$; \smallskip \item{$\bullet$} $L(\X,\Y)$, resp. $\tilde L(\X,\Y)$, denotes the space of linear, resp. conjugate linear, operators from the Banach space $\X$ to the Banach space $\Y$. \smallskip \item{$\bullet$} $B(\X,\Y)$, resp. $\tilde B(\X,\Y)$, denotes the space of bounded, everywhere defined, linear, resp. conjugate linear, operators on the Banach space $\X$ to the Banach space $\Y$. It is a Banach space with the norm $\|A\|_{\X,\Y}:=\sup\{\|Ax\|_\Y,\, \|x\|_\X=1\}$. %\smallskip %\item{$\bullet$} $\CO(\X,\Y)$, resp. $\tilde\CO(\X,\Y)$, denotes the %space of linear, resp. conjugate linear, closed operators %between the Banach spaces $\X$ and $\Y$. \smallskip \item{$\bullet$} The closed linear operator operator $A'$ and the conjugate linear closed operator $\tilde A'$ are the adjoints of the densely defined linear operator $A$ and of the densely defined conjugate linear operator $\tilde A$ respectively, i.e. \p \centerline{\text {$\forall\,x\in D(A)\subseteq\X,\quad\forall \tilde\ell\in D(A')\subseteq Y',\qquad (A'\ell)(x)=\ell(Ax)$,}}\p \centerline{\text {$\forall\,x\in D(\tilde A)\subseteq\X,\quad \forall\tilde\ell\in D(\tilde A')\subseteq Y',\qquad(\tilde A'\ell)(x)=(\,\ell(\tilde Ax)\,)^*$,}}\par where $^*$ denotes complex conjugation. \smallskip \item{$\bullet$} $\tilde S(\X',\X)$ denotes the space of conjugate linear operators $A$ such that \p \centerline{\text{$\forall\,\ell_1,\ell_2\in D(A),\qquad \ell_1(A\ell_2)=(\,\ell_2(A\ell_1)\,)^*$. }} \smallskip \item{$\bullet$} For any $A\in\tilde S(\X',\X)$ we define $\gamma(A):= \inf\left\{\,\ell(A \ell),\ \|\ell\|_{\X'}=1,\ \ell\in D(A)\,\right\}$. \smallskip \item{$\bullet$} $J_\X\in B(\X,\X'')$ indicates the injective map (an isomorphism when $\X$ is reflexive) defined by $(J_\X x)(\ell):=\ell(x)$. %$x\in \X$, $\ell\in \X'$. \smallskip \item{$\bullet$} If $\H$ is a complex Hilbert space with scalar product (conjugate linear w.r.t. the first variable) $\langle\cdot,\cdot\rangle$, then $C_\H\in\tilde B(\H,\H')$ denotes the isomorphism defined by $(C_\H y)(x):=\langle y,x\rangle$. The Hilbert adjoint of the densely defined linear operator $A$ is then given by $A^*= C_\H^{-1}\cdot A'\cdot C_\H$. %Where the conjugate linearity of $C_\H$ plays no role we identify %$\H$ with its dual $\H'$ without further specifications. \smallskip \item{$\bullet$} $\F$ and $*$ denote Fourier transform and convolution respectively. \smallskip \item{$\bullet$} $H^s(\RE^n)$, $s\in\RE$, is the usual scale of Sobolev--Hilbert spaces, i.e. $H^s(\RE^n)$ is the space of tempered distributions with a Fourier transform which is square integrable w.r.t. the measure with density $(1+|x|^2)^s$. \smallskip \item{$\bullet$} $c$ denotes a generic strictly positive constant which can change from line to line. \vskip 30pt\p %\vfill\eject {\bf 2. A KREIN--LIKE FORMULA} \vskip 10pt\p Let $$A:D(A)\to\H$$ be a self-adjoint operator on the complex Hilbert space $\H$. $D(A)$ inherits a Hilbert space structure by introducing the usual scalar product leading to the graph norm $\|\phi\|^2_A:=\langle\phi,\phi\rangle+\langle A\phi,A\phi\rangle$. Denoting the resolvent set of $A$ by $\rho(A)$ we define, for any $z\in\rho(A)$, $$ R^0(z):=(-A+z)^{-1}:\H\to D(A)\,,\qquad R^0(z)\in B(\H,D(A))\ . $$ %We will retain the notation $(H+z)^{-1}$ when treating the resolvent %of $H$ as an operator in $B(\H)$. \par We consider now a linear operator $$ \tau:D(A)\to\X \,,\qquad \tau\in B(D(A),\X)\ , $$ where $\X$ is a complex Banach space. By means of $A$ and $\tau$ we can define, for any $z\in\rho(A)$, the following operators: $$ \breve G(z):=\tau\cdot R^0(z) :\H\to\X\,,\qquad\breve G(z)\in B(\H,\X)\ , %\eqno(1) $$ $$ G(z):=C_\H^{-1}\cdot\breve G(z^*)':\X'\to\H\,,\qquad G(z)\in\tilde B(\X',\H)\ . %\eqno(2) $$ \v {\bf Remark 1.} Being $R^0(z)$ surjective, $R^0(z)'$ is injective. If $\tau$ has dense range then $\tau'$ is injective. Therefore, when $\tau$ has dense range, $\breve G(z)$ is surjective and $G(z)$ is injective. This implies that the only $\Lambda\in B(\X,\X')$ which solves the operator equation $G(z)\cdot\Lambda\cdot \breve G(z)=0$ is the zero operator. \v {\bf Lemma 1.} $$ (z-w)\,\breve G(w)\cdot R^0(z)=\breve G(w)-\breve G(z) $$ $$ (z-w)\, R^0(w)\cdot G(z)=G(w)-G(z)\ . $$ \v {\bf Remark 2.} The second relation in the lemma above shows that $$ \forall\,w,z\in\rho(A),\ w\not= z,\qquad \text{\rm Range}(\,G(w)-G(z)\,)\subseteq D(A)\ . $$ \v {\it Proof of lemma 1.} By first resolvent identity one has $$(z-w)\,R^0(w)\cdot R^0(z)=R^0(w)-R^0(z)\ . $$ Therefore $$\eqalign{ (z-w)\,\breve G(w)\cdot R^0(z)=& (z-w)\,\tau\cdot R^0(w)\cdot R^0(z)\cr =& \breve G(w)-\breve G(z) } $$ and, by duality (here $R^0(z)$ is considered as an element of $B(\H,\H)$), $$\eqalign{ &G(w)-G(z)=C_\H^{-1}\cdot\left(\breve G(w^*)-\breve G(z^*)\right)'\cr =&C_\H^{-1}\cdot\left((z^*-w^*)\,\breve G(z^*)\cdot R^0(w^*)\right)'\cr =&C_\H^{-1}\cdot\left((z^*-w^*)\,C_\H\cdot R^0(w)\cdot C_\H^{-1}\cdot \breve G(z^*)' \right)\cr =&(z-w)\, R^0(w)\cdot G(z) \ .} $$\qed\v We want now to define a new self-adjoint operator which, when restricted to the kernel of $\tau$, coincides with the original $A$. Since, in the case of a bounded perturbation $V$, one has, for any $z\in\C\backslash\RE$ such that $\|V\cdot R^0(z)\|_{\H,\H}<1$, $$ (-(A+V)+z)^{-1}=R^0(z)+ R^0(z)\cdot(\,\uno-V\cdot R^0(z)\,)^{-1}\cdot V\cdot R^0(z)\ , $$ we are lead to write the presumed resolvent as $$ R^\tau(z)=R^0(z)+B(z)\cdot\tau\cdot R^0(z)\equiv R^0(z)+B(z)\cdot\breve G(z)\,, $$ where $B(z)\in B(\X,\H)$ has to be determined. \par Self-adjointness requires $R^\tau(z)^*=R^\tau(z^*)$ or, equivalently, $$ G(z)\cdot B(z^*)'\cdot C_\H=B(z)\cdot\breve G(z)\ .\eqno(1) $$ Therefore if we put $B(z)=G(z)\cdot \Lambda(z)$, $\Lambda(z)\in\tilde B(\X,\X')$, then one can check that (1) is implied by (by Remark 1, when $\tau$ has dense range, is equivalent to) $$ \Lambda(z)'\cdot J_\X=\Lambda(z^*)\ .\eqno(2) $$ We now impose the resolvent identity $$ (z-w)\,R^\tau(w)R^\tau(z)=R^\tau(w)-R^\tau(z)\ .\eqno(3)$$ Since (we make use of lemma 1) $$ \eqalign{ &(z-w)\,R^\tau(w)\cdot R^\tau(z)=(z-w)\, \left(R^0(w)\cdot R^0(z)\right.\cr +& R^0(w)\cdot G(z)\cdot\Lambda(z)\cdot\breve G(z) +G(w)\cdot\Lambda(w)\cdot\breve G(w)\cdot R^0(z)\cr +&\left.G(w)\cdot\Lambda(w)\cdot\breve G(w)\cdot G(z)\cdot\Lambda(z)\cdot\breve G(z)\right)\cr =& R^0(w)- R^0(z) +G(w)\cdot\Lambda(z)\cdot\breve G(z) -G(z)\cdot\Lambda(z)\cdot\breve G(z)\cr +&G(w)\cdot\Lambda(w)\cdot\breve G(w) -G^0(w)\cdot\Lambda(w)\cdot\breve G(z)\cr +&(z-w)\,G(w)\cdot\Lambda(w)\cdot\breve G(w)\cdot G(z)\cdot\Lambda(z)\cdot\breve G(z)\cr =&R^\tau(w)-R^\tau(z) +G(w)\cdot\left( \Lambda(z)-\Lambda(w)\right)\cdot\breve G(z)\cr +&(z-w)\,G(w)\cdot\Lambda(w)\cdot\breve G(w)\cdot G(z)\cdot\Lambda(z)\cdot\breve G(z)\ ,} $$ the relation (3) is implied by (by Remark 1, when $\tau$ has dense range, is equivalent to) $$ \Lambda(w)-\Lambda(z)= (z-w)\,\Lambda(w)\cdot\breve G(w)\cdot G(z)\cdot\Lambda(z)\ .\eqno(4) $$ Suppose now that there exists a (necessarily closed) operator $$\Gamma(z):D\subseteq\X'\to\X $$ such that, for some open set $Z\subseteq\rho(A)$ such that $z\in Z$ iff $z^*\in Z$, one has $$ \forall\, z\in Z,\qquad\Gamma(z)^{-1}=\Lambda(z)\ . $$ Then we have that (4) forces $\Gamma(z)$ to satisfy the relation $$ \Gamma(z)-\Gamma(w)=(z-w)\,\breve G(w)\cdot G(z)\ .\eqno(5) $$ which is equivalent to $$ \forall\, \ell\in D\subseteq\X',\qquad \d \Gamma(z)\ell=\breve G(z)\cdot G(z)\ell\ .\eqno(6) $$ Regarding the identity (2), suppose that $$ \forall\,\ell_1,\ell_2\in D,\qquad \ell_1(\Gamma(z^*)\ell_2)=(\,\ell_2(\Gamma(z)\ell_1)\,)^*\ .\eqno(7) $$ This, in the case $\Gamma(z)$ has a bounded inverse given by $\Lambda(z)$ as we are pretending, implies (2) which, if $\Gamma(z)$ is densely defined, is then equivalent (use e.g. [K, thm. 5.30, chap. III]) to $$\Gamma(z)'=J_\X\cdot \Gamma(z^*)\ . \eqno(7.1)$$ We will therefore concentrate now on the set of maps $$ \Gamma:\rho(A)\to \tilde L(\X',\X) $$ which satisfy (6) (equivalently (5)) and (7) (we are implicitly supposing that $D$, the domain of $\Gamma(z)$, is $z$--independent).\par An explicit representation of the set of such maps is given by the following \v {\bf Lemma 2.} {\it Given any $z_0\in \rho(A)$ the map $$ \hat\Gamma:\rho(A)\to \tilde B(\X',\X)\qquad \hat\Gamma(z):=\tau \cdot\left(\,\frac{G(z_0)+G(z_0^*)}{2}-G(z)\,\right)\eqno(8) $$ satisfies (6) and (7).} \v {\bf Remark 3.} Lemma 2 shows that the set of maps $$ \Gamma:\rho(A)\to \tilde L(\X',\X) $$ which satisfy (6) and (7) can be parametrized by $\tilde S(\X',\X)$. Indeed, by (6), any of such maps must differ from $\hat\Gamma(z)\in \tilde B(\X',\X)$ by a $z$--independent operator in $\tilde S(\X',\X)$. Therefore any parametrization is of the kind $$ \Gamma_\Theta:\rho(A)\to \tilde L(\X',\X)\qquad \Gamma_\Theta(z)=\Theta+\Gamma(z)\,,\quad \Theta\in \tilde S(\X',\X)\ ,\eqno(9) $$ where $\Gamma(z)$ is some map which satisfy (6) and (7). \v {\it Proof of lemma 2.} By lemma 1 one has $$ (z-w)\,\breve G(w)\cdot G(z)=\tau(\,G(w)-G(z)\,)= \tau(\,G(z_0)-G(z)\,)-\tau(\,G(z_0)-G(w)\,) $$ and so $\tau \cdot( G(z_0)-G(z))$ solves (5); by linearity also $\hat\Gamma(z)$ is a solution.\par As regard (7) let us at first note that $$ J_\X\cdot\breve G(z)=\breve G(z)''\cdot J_\H $$ and $$ \left(C_\H'\cdot J_\H y\right)(x)=\left(J_\H y\left(C_H x\right)\right)^* =\left(C_\H x(y)\right)^*=\langle x,y\rangle^*=C_\H y(x)\ . $$ Therefore one has $$ \eqalign{ &\left(\breve G(w)\cdot G(z)\right)'=G(z)'\cdot\breve G(w)'= \breve G(z^*)''\cdot\left(C_\H'\right)^{-1}\cdot\breve G(w)'\cr =&J_\X\cdot\breve G(z^*)\cdot J_\H^{-1}\cdot\left(C_\H'\right)^{-1}\cdot\breve G(w)' =J_\X\cdot\breve G(z^*)\cdot C_\H^{-1}\cdot\breve G(w)'\cr =&J_\X\cdot\breve G(z^*)\cdot G(w^*) } $$ which immediately implies that $\hat\Gamma(z)$ satisfies (7). \qed \v Lemma 2 does not entirely solve the problem of the search of $\Gamma(z)$ since $\hat\Gamma(z)$ can give rise to non-local boundary conditions (see Remark 7 below); moreover $\hat \Gamma(z)$ explicitly depends on the choice of a particular $z_0\in\rho(A)$. However the boundedness of $\hat\Gamma(z)$ implies a useful criterion for obtaining other maps $\Gamma(z)$ which satisfy (6) and (7):\v {\bf Lemma 3.} {\it Suppose that $$\tilde \Gamma(z):D(\tilde\Gamma) \subseteq\X'\to\X\,,\qquad z\in\rho(A)\,, $$ is a family of conjugate linear, densely defined operators such that $$ \forall\,\ell_1,\ell_2\in D(\tilde\Gamma)\,,\qquad \ell_2(\tilde\Gamma(z^*)\ell_1)=(\,\ell_1(\tilde\Gamma(z)\ell_2)\,)^*\eqno(10) $$ and $$ \forall\, \ell\in E,\, \forall\,\ell_1\in D(\tilde\Gamma)\,, \qquad\d \ell(\tilde\Gamma(z)\ell_1)=\ell (\breve G(z)\cdot G(z)\ell_1)\equiv \langle G(z^*)\ell,G(z)\ell_1\rangle\ ,\eqno(11) $$ where $E\subseteq\X'$ is either a dense subspace or the dual of some Schauder base in $\X$. Then $\tilde\Gamma(z)$ is closable and its closure satisfies (6) and (7). } \proof By (11) necessarily $\tilde\Gamma(z)$ differs from (the restriction to $D(\tilde\Gamma)$ of) $\hat\Gamma(z)$ by a $z$--independent, densely defined operator $\tilde\Theta\in\tilde S(\X',\X)$. Being densely defined, $\tilde\Theta$ has an adjoint and $\tilde\Theta\subseteq\Theta'$. Therefore $\tilde\Theta$ is closable and so, being $\hat\Gamma(z)$ bounded, $\tilde\Gamma(z)=\tilde\Theta+\hat\Gamma(z)$ is closable. Denoting by $\Theta$ the closure of $\tilde\Theta$, the closure of $\tilde \Gamma(z)$ is given by $\Theta+\hat\Gamma(z)$, which satisfies (6) and (7) by lemma 2.\qed \v We can state now our main result: \v {\bf Theorem 1.} {\it Let $\Gamma_\Theta(z)$ as in (9). Under the hypotheses $$ Z_\Theta:=\left\{z\in\rho(A)\ :\ \exists\, \Gamma_\Theta(z)^{-1}\in\tilde B(\X,\X'),\ \exists\, \Gamma_\Theta(z^*)^{-1}\in\tilde B(\X,\X')\right\}\,\not=\,\emptyset\eqno(\text{\rm h1}) $$ $$ \text{\rm Range$\,(\tau')\cap \H'=\left\{0\right\}$}\, ,\eqno(\text{\rm h2}) $$ the bounded linear operator $$ R^\tau_\Theta(z):=R^0(z)+G(z)\cdot\Gamma_\Theta(z)^{-1}\cdot\breve G(z)\,,\qquad z\in Z_\Theta\, , $$ is a resolvent of the self-adjoint operator $A^\tau_\Theta$ which coincides with $A$ on the kernel of $\tau$ and is defined by $$ D(A_\Theta^\tau):=\left\{\,\phi\in\H\ :\ \phi= \phi_z+G(z)\cdot\Gamma_\Theta(z)^{-1}\cdot\tau\,\phi_z,\quad \phi_z\in D(A)\,\right\}\, , $$ $$ (-A_\Theta^\tau+z)\phi:=(-A+z)\phi_z\ . $$ Such a definition is $z$--independent and the decomposition of $\phi$ entering in the definition of the domain is unique. } \v {\bf Remark 4.} Viewing $A$ as a bounded operator on $D(A)$ to $\H$, we can consider the adjoint $(-A+z^*)'$, so that $$ (-A+z^*)'\cdot C_\H :\H\to D(A)'\,,\qquad (-A+z^*)'\cdot C_\H{\,} _{\left| D(A)\right.}= C_\H\cdot(-A+z) $$ and, by the definition of $G(z)$, $$ (-A+z^*)'\cdot C_\H \cdot G(z)=\tau'\ . $$ Therefore, defining $Q_\phi:=\Gamma_\Theta(z)^{-1}\cdot\tau\,\phi_z$, one has $$ C_\H\cdot(-A_\Theta^\tau+z)\phi=(-A+z^*)'\cdot C_\H\phi_z =(-A+z^*)'\cdot C_\H\phi-\tau'Q_\phi\ , $$ i.e. $$ A^\tau_\Theta\phi=C_\H^{-1}\cdot\left(A'\cdot C_\H\phi+\tau'Q_\phi\right)\ . $$ Formally re-writing the last relation as $$ A^\tau_\Theta\phi=A\phi+C_\H^{-1}\cdot\tau'Q_\phi\ , $$ we can view $A^\tau_\Theta$ as a perturbation of $A$, the perturbation being singular since, by (h2), $\tau' Q_\phi\in D(A)'\backslash \H'$. \v {\bf Remark 5.} 1. If $\X$ is reflexive and $\Gamma_\Theta(z)$ is densely defined, then, by (7.1), there follows $$ \Gamma_\Theta(z)^{-1}\in\tilde B(\X,\X')\quad \Longrightarrow\quad \Gamma_\Theta(z^*)^{-1}\in\tilde B(\X,\X')\ . $$ \p 2. If $Z_\Theta\not=\emptyset$ then $Z_\Theta$ is necessarily open. Indeed, by (5), $$\Gamma_\Theta(z+h)=\Gamma_\Theta(z)+h\,\breve G(z)\cdot G(z+h)\ ,$$ and so $\Gamma_\Theta(z+h)^{-1}\in\tilde B(\X',\X)$ if $z\in Z_\Theta$ and $h$ is sufficiently small.\p 3. If in the representation (9) there exists $z_0\in\rho(A)$ such that $\Gamma(z_0)=\Gamma(z_0^*)=0$ (this is certainly true if $\rho(A)\cap\RE\not=\emptyset$ and if one uses representation (8) with $z_0\in\RE$) then obviously $Z_\Theta$ is non-empty for any invertible $\Theta\in\tilde S(\X',\X)$. A more significative criterion leading to (h1) will be given in Proposition 1 below. \v {\bf Remark 6.} 1. By the definition of $G(z)$ one has that (h2) is equivalent to $$ \text{\rm Range$\,(\,G(z)\,)\cap D(A)=\left\{0\right\}$}\ .\eqno(\text{\rm h2}) $$ 2. If Kernel$\,(\tau)$ is dense then (h2) holds true (it is not hard to show that the reverse implication is false). Indeed the density hypothesis implies, if $Q\in\X'$, $$ \forall\,\psi\in\text{\rm Kernel$\,(\tau)$},\qquad\langle\phi,\psi\rangle=Q(\tau\psi) =\tau'Q(\psi) \quad\Longrightarrow \quad\phi=0\ . $$ This, by the definition of $G(z)$, implies $$ R^0(z)\phi= G(z)Q\quad\Longrightarrow \quad\phi=0\ , $$ which gives (h2). \v {\bf Remark 7.} If in the above theorem one uses the representation $\hat\Gamma_\Theta(z):=\Theta+\hat\Gamma(z)$ given by lemma 2 and Range$\,\tau=\X$, then one can readily check that the domain of $A^\tau_\Theta$ is equivalently characterized in term of ``generalized boundary conditions'': $\phi\in D(A^\tau_\Theta)$ if and only if $$ \exists\,Q_\phi\in D(\Theta)\subseteq\X'\quad\text{\rm such that}\quad \phi-G(z)Q_\phi\in D(A) $$ and $$ \tau\left(\phi-\,\frac{G(z_0)+G(z_0^*)}{2}\,Q_\phi\right)=\Theta\, Q_\phi\ . $$ If $\Y=$Range$\,\tau\subsetneq\X$ then one can not explicitly characterize the regularity of the ``charges'' $Q_\phi$. Indeed in this case one only knows that $Q_\phi\in \Gamma_\Theta(z)^{-1}(\Y)$. \v {\it Proof of theorem 1.} We have already proven that, under our hypotheses, $R_\Theta^\tau(z)$ is a pseudo-resolvent, i.e. $$ (z-w)\,R^\tau_\Theta(w)R^\tau_\Theta(z)=R^\tau_\Theta(w)-R^\tau_\Theta(z)\ .\eqno(12) $$ We proceed now as in the proof of [AGHH, thm. II.1.1.1]. By [K, chap. VIII, \S 1.1] $R^\tau_\Theta(z)$, being a pseudo-resolvent, is the resolvent of a closed operator if and only if it is injective. Since $R^\tau_\Theta(z)\phi=0$ would imply $$ R^0(z)\phi=-G(z)\cdot\Gamma_{\Theta}(z)^{-1}\cdot\breve G(z)\phi\ , $$ by (h2) we have $R^0(z)\phi=0$ and so $\phi=0$. \par Since, as we have seen before, (7) implies, when $z\in Z_\Theta$, $$ \Gamma_\Theta(z^*)^{-1}=\left(\Gamma_\Theta(z)^{-1}\right)'\cdot J_\X\,, $$ one has $$\eqalign{ &\left(G(z)\cdot\Gamma_\Theta(z)^{-1}\cdot\breve G(z)\right)^*\cr =&C_\H^{-1}\cdot\left(G(z)\cdot\Gamma_\Theta(z)^{-1}\cdot\breve G(z)\right)'\cdot C_\H\cr =&C_\H^{-1}\cdot\breve G(z)'\cdot\left(\Gamma_\Theta(z)^{-1}\right)'\cdot\breve G(z^*)''\cdot (C_\H')^{-1}\cdot C_\H\cr =&G(z^*)\cdot\left(\Gamma_\Theta(z)'\right)^{-1}\cdot J_\X\cdot\breve G(z^*)\cdot J_\H^{-1}\cdot(C_\H')^{-1}\cdot C_\H\cr =&G(z^*)\cdot\Gamma_\Theta(z^*)^{-1}\cdot\breve G(z^*)\,, } $$ and so $$R_\Theta^\tau(z)^*=R_\Theta^\tau( z^*)\ .$$ This gives the denseness of $D(A_\Theta^\tau):=\text{\rm Range}(R^\tau_\Theta(z))$. Indeed $\phi\perp D(A_\Theta^\tau)$, which is equivalent to $\langle R_\Theta^\tau( z^*)\phi,\psi\rangle=0$ for all $\psi\in\H$, implies $\phi=0$. \par Let us now define, on the dense domain $D(A_\Theta^\tau)$, the closed operator $$ A_\Theta^\tau:=R_\Theta^\tau(z)^{-1}-z $$ which, by the resolvent identity (12), is independent of $z$; it is self-adjoint since $$ ((A_\Theta^\tau)^*+ z^*)^{-1}=R_\Theta^\tau(z)^*=R_\Theta^\tau( z^*)=(A_\Theta^\tau+ z^*)^{-1}\ . $$ To conclude, the uniqueness of the decomposition $$ \phi= \phi_z+G(z)\cdot\Gamma_\Theta(z)^{-1}\cdot\tau\,\phi_z\,, \qquad \phi\in D(A^\tau_\Theta)\,, $$ is an immediate conseguence of (h2). \qed \v The following result states that (h1) holds true under relatively weak hypotheses: \v {\bf Proposition 1.} {\it Suppose that $\tau$ has dense range, and let $\Gamma_\Theta(z)$ as in (9) be closed and densely defined. Then one has $$ \C\backslash\RE\cup W^-_\Theta\cup W_\Theta^+\subseteq Z_\Theta\,, $$ where $$ W^\pm_\Theta:=\left\{\,\lambda\in\RE\cap \rho(A)\ :\ \gamma(\pm\Gamma(\lambda))>-\gamma(\pm\Theta)\,\right\}\ . $$ } \v Since $Z_\Theta\subseteq\rho(A^\tau_\Theta)$, the above proposition implies a semi-boundedness criterion for the extensions $A_\Theta^\tau$: \v {\bf Corollary 1.} {\it Let $-A$ be bounded from below and suppose that there exist $\lambda_0 \in\rho(A)$ and $\theta_0\in\RE$ such that $$ \forall\,\lambda\ge\lambda_0\qquad \gamma(\Gamma(\lambda))>-\theta_0\ . $$ Then $$\inf\sigma(-A_\Theta^\tau)\ge-\lambda_0 $$ for any $\Theta\in\tilde S(\X',\X)$ such that $\gamma(\Theta)\ge \theta_0$. } \v {\it Proof of Proposition 1.} Since $\Theta\in\tilde S(\X',\X)$ implies $\ell(\Theta\ell)\in\RE$ and $$ \eqalign{ \ell(\Gamma(z)\ell)-\left(\ell(\Gamma(z)\ell)\right)^* =&\ell(\Gamma(z)\ell)-\Gamma(z)'\ell(\ell)= \ell(\Gamma(z)\ell)-J_\X\cdot \Gamma(z^*)\ell(\ell)\cr =&\ell((\Gamma(z)-\Gamma(z^*))\ell)=(z-z^*)\ell(\breve G(z^*)\cdot G(z)\ell)\cr =&(z-z^*)\|G(z)\ell\|_\H^2\ , } $$ one has $$ \Gamma_\Theta(z)\ell=0,\ z\in\C\backslash\RE\quad\Rightarrow\quad \text{\rm Im}(z)\|G(z)\ell\|_\H^2=0 $$ and $$ \Gamma_\Theta(\lambda)\ell=0,\ \lambda\in\RE\cap\rho(A)\quad\Rightarrow\quad -\ell(\Theta\ell)=\ell(\Gamma(\lambda)\ell)\ . $$ Injectivity of $\Gamma_\Theta(z)$ then follows by injectivity of $G(z)$ (see Remark 1) and the definitions of $W^\pm_\Theta$.\p By Hahn--Banach theorem the range of $\Gamma_\Theta(z)$ is dense if and only if its annihilator is equal to $\left\{0\right\}$, i.e. if and only if $$ \left\{\ell_2\in\X'\ :\ \forall\,\ell_1\in D(\Gamma_\Theta)\quad \ell_2(\Gamma_\Theta(z)\ell_1)=0\right\} =\left\{0\right\}\ . $$ Being $D(\Gamma_\Theta)$ dense this is equivalent to $$ \left\{\ell_2\in D(\Gamma_\Theta)\ :\ \forall\,\ell_1\in D(\Gamma_\Theta)\quad \ell_2(\Gamma_\Theta(z)\ell_1)=0\right\} =\left\{0\right\}\ , $$ and so relation (7) and injectivity give denseness of the range of $\Gamma_\Theta(z)$. By open mapping theorem, to conclude the proof we need to prove that the range of $\Gamma_\Theta(z)$ is closed. For any $\ell\in D(\Gamma_\Theta)$, $\|\ell\|_{\X'}=1$, one has $$ \|\Gamma_\Theta(z)\ell \|_\X^2\ge |\ell(\Gamma_\Theta(z)\ell)|^2= \left(\ell(\Theta \ell)+\re\left(\ell(\Gamma(z)\ell)\right)\right)^2+ \text{\rm Im}(z)^2\|G(z)\ell\|_\H^4\ . $$ Therefore, when $z\in\C\backslash\RE\cup W^-_\Theta\cup W^+_\Theta$, one has $$ \inf\left\{\,\|\Gamma_\Theta(z)\ell \|_\X\,,\ \|\ell\|_{\X'}=1\,\right\}>0\ , $$ and so, since $\Gamma_\Theta(z)$ is closed, it has a closed range by [K, thm. 5.2, chap. IV]. \qed \v {\bf Remark 8.} By the proposition above, if $\X=$Range$\,(\tau)$ is finite-dimensional and $\Gamma_\Theta(z)$ is everywhere defined, then (h1) is satisfied with at least $\C\backslash\RE\subseteq Z_\Theta$. \v {\bf Remark 9.} By the proposition above, since $\hat \Gamma(z)$ is bounded, if one uses the representation $\hat \Gamma_\Theta(z)$, with $\Theta\in \tilde S(\X',\X)$ closed and densely defined, then (h1) is satisfied (with at least $\C\backslash\RE\subseteq Z_\Theta$) when $\tau$ has dense range. \v {\bf Remark 10.} If $\X$ is a Hilbert space (with scalar product $\langle\cdot,\cdot\rangle$) we can of course use the map $C_\X$ to identify $\X$ with $\X'$ and re-define $G(z)$ as $$ G(z):=C_\H^{-1}\cdot\breve G(z^*)'\cdot C_\X:\X\to\H\ . $$ The statements in the above theorem remain then unchanged taking $$ \Gamma_\Theta:\rho(A)\to L(\X,\X)\,,\qquad \Gamma_\Theta(z)=\Theta+\Gamma(z) $$ with $\Theta$ such that $$ \forall\, x,y\in D(\Theta),\qquad \langle \Theta x,y\rangle= \langle x,\Theta y\rangle $$ and $\Gamma(z)$ satisfying (6) and $$ \forall\, x,y\in D(\Gamma),\qquad \langle \Gamma(z)x,y\rangle=\langle x,\Gamma(z^*)y\rangle\ . $$ {\bf Remark 11.} When $\X$ is a Hilbert space, by Theorem 1, since $G(z)$ and $\breve G(z)$ are bounded, we have that $R^\tau_\Theta(z)-R^0(z)$ is a trace class operator on $\H$ if and only if $(\Theta+\Gamma(z))^{-1}$ is a trace class operator on $\X$ (see e.g. [K, \S 1.3, chap. X]). This information can be used (proceeding along the same lines as in [BT]) to infer from $\sigma(A)$ some properties of $\sigma(A^\tau_\Theta)$. \v In the Hilbert space case one can give, beside the one appearing in Remark 4, another criterion for obtaining the map $\Gamma_\Theta$. Indeed one has the following \v {\bf Lemma 4. } {\it Suppose that there exists a densely defined sesquilinear form $\tilde\E(z)$, $z\in\rho(A)$, with $z$--independent domain $D(\tilde\E)\times D(\tilde\E)$, such that $$ \forall\,x,y\in D(\tilde\E),\qquad\tilde\E(z^*)(x,y)=(\,\tilde\E(z)(y,x)\,)^*\ ,\eqno(13) $$ $$ \forall\,x,y\in D(\tilde\E),\qquad\d\tilde\E(z)(x,y)=\langle G(z^*)x,G(z)y\rangle\ , \eqno(14) $$ and such that there exist $z_0\in\rho(A)$, $M\in\RE$ for which $\tilde\E(z_0)$ is closable and $$ \forall\,x\in D(\tilde\E),\qquad\text{\rm Re}\left(\tilde\E(z_0)(x,x)\right) \ge M\,\langle x,x\rangle\ .\eqno(15) $$ Then $\tilde\E(z)$ is closable for any $z\in\rho(A)$ and, denoting by $\E(z)$ it closure, there exists a densely defined, closed linear operator $\Gamma(z)$ with z--independent domain $D(\Gamma)$, defined by $$ \forall\,x\in D(\E),\,\forall\,y\in D(\Gamma),\qquad\E(z)(x,y) =\langle x,\Gamma(z)y\rangle\ ,$$ satisfying (6) and the Hilbert space analogue of (7.1), i.e. $$\Gamma(z)^* =\Gamma(z^*)\ .$$} \proof By our hypotheses $\tilde\E(z)$ necessarily differs from (the restriction to $D(\tilde\E)\times D(\tilde\E)$ of) the bounded sesquilinear form associated to $\hat\Gamma(z)$ by a $z$--independent Hermitean form $\tilde\Cal Q$. Therefore $$\tilde\Q(x,y)=\tilde\E(z_0)(x,y)-\langle x,\hat\Gamma(z_0)y\rangle$$ is a semi-bounded, densely defined, closable Hermitean form. If $\Theta$ denotes the unique semi-bounded self-adjoint operator corresponding to the closure of $\tilde\Q$ (the existence of $\Theta$ is guaranteed by [K, thm. 2.6, chap. VI]), then the operator $\Gamma(z):=\Theta+\hat\Gamma(z)$ gives the thesis. \qed \vskip 30pt\p {\bf 3. APPLICATIONS} \vskip 10pt\p {\bf Example 1.} {\it The $\H_{-2}$--construction.}\v Let $\X=\C$, $\varphi\in D(A)'\backslash\left\{0\right\}$ and put $\tau=\varphi$. Defining $$ \tilde R^0(z):=C_\H^{-1}\cdot R^0(z^*)' \in B(D(A)',\H)$$ one has then $$ \breve G(z):\H\to\C\,,\qquad\breve G(z)\phi=\langle\tilde R^0(z^*)\varphi,\phi\rangle $$ and $$ G(z):\C\to\H\,,\qquad G(z)\zeta=\zeta\tilde R^0(z)\varphi\ . $$ The hypotheses (h2) is equivalent to the request $$ \varphi\notin \H'\ , $$ whereas hypotheses (h1) is always satisfied with at least $\C\backslash\RE\subseteq Z_\Theta$ since $\X$ is finite dimensional (see Remark 8). Then the self-adjoint operator $A^\varphi_\alpha$ has resolvent $$ (-A^\varphi_\alpha+z)=(-A+z)^{-1}+\Gamma_\alpha(z)^{-1}\tilde R^0(z)\varphi\otimes \tilde R^0(z^*)\varphi\ , %\qquad z\in\C\backslash\RE\ , $$ where (by lemma 2) $$ \Gamma_\alpha(z) =\alpha+\varphi\left(\frac{\tilde R^0(z_0)\varphi+\tilde R^0(z_0^*)\varphi}{2}- \tilde R^0(z)\varphi\right)\,,\qquad\alpha\in\RE\ . $$ This coincides with the $``\,\H_{-2}$--construction'' given in [KS] (there only the case $-A\ge 0$, $z_0=1$ was considered). For a similar construction also see [AK] and references therein. \v\v {\bf Example 2.} {\it A variation on the Birman--Kre\u\i n--Vishik theory.} \v Let $A$ be a strictly positive self-adjoint operator, so that $0\in \rho(-A)$, and let $\tau: D(A)\to\X$ satisfy (h2). By Remark 5.3 and theorem 1, for any $\Theta\in\tilde S(\X',\X)$ which has a bounded inverse, we can define the (strictly positive when $\Theta$ is positive, i.e. $\gamma(\Theta)\ge 0$) self-adjoint opertor $A^\tau_\Theta$ by $$ ({A^\tau_\Theta})^{-1}=A^{-1}+G\cdot\Theta^{-1}\cdot\breve G\ , $$ where $G:=G(0)$ and $\breve G:=\breve G(0)$. This gives a variation of the Birman--Kre\u\i n--Vishik approach which comprises the result given in [KKO]. In particular [KKO, example 4.1] can be obtained by taking $\H=L^2(\Omega)$, $A=-\Delta_\Omega+\lambda$, $\lambda>0$, $\Omega=(0,\pi)\times\RE^2$, $D(\Delta_\Omega)=H^2_0(\Omega)$, $\tau:H^2_0(\Omega)\to L^2(0,\pi)$ the evaluation along the segment $\left\{(x,0,0), x\in (0,\pi)\right\}$, $\Theta= -\Delta_{(0,\pi)}$, $D(\Theta)=H^2_0(0,\pi)$; [KKO, example 4.2] corresponds to $\H=L^2(\RE^3)$, $A=-\Delta+\lambda$, $\lambda>0$, $D(\Delta)=H^2(\RE^3)$, whereas $\tau$ and $\Theta$ are the same as before. \v\v {\bf Example 3.} {\it Finitely many point interactions in three dimensions.}\v We take $\H=L^2(\RE^3)$, $A=\Delta$, $D(A)=H^2(\RE^3)\subset C_b(\RE^3)$. Considering then a finite set $Y\subset\RE^3$, $\#Y=n$, we take as the linear operator $\tau$ the linear continuous surjective map $$ \tau_Y:H^2(\RE^3)\to\C^n\qquad\tau_Y\phi:=\{\phi(y)\}_{y\in Y}\ . $$ Then one has $$ \breve G(z):\LD\to\C^n\,,\qquad\breve G(z)\phi=\{\G_z*\phi\,(y)\}_{y\in Y}\ , $$ where $$\G_z(x)=\frac{e^{-\sqrt z|x|}}{4\pi|x|}\,,\qquad \re\sqrt z>0\,,\qquad\G_z^y(x):=\G_z(x-y)\ ,$$ and $$ G(z):\C^n\to\LD\,,\qquad G(z)\zeta=\sum_{y\in Y}\zeta_y\G_z^y \equiv \G_z*\sum_{y\in Y}\zeta_y\delta_y\ . $$ A straightforward calculation then gives $$\eqalign{ &\left(\breve G(z)\cdot G(z)\zeta\right)_y =\sum_{\tilde y\in Y}\zeta_{\tilde y} \langle\G^{\tilde y}_{z^*},\G^y_z\rangle\cr =&\zeta_y\,\frac{1}{(2\pi)^3} \int_{\RE^3} dk\, \frac{1}{(|k|^2+z)^2} +\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\frac{1}{(2\pi)^3} \int_{\RE^3} dk\, \frac{e^{-ik\cdot(\tilde y-y)}}{(|k|^2+z)^2}\cr =&\zeta_y\,\frac{1}{2\pi^2} \int_0^\infty dr\, \frac{r^2}{(r^2+z)^2} +\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\frac{1}{2\pi^2|\tilde y-y|} \int_0^\infty dr\, \frac{r\sin(r|\tilde y-y|)}{(r^2+z)^2}\cr =&\zeta_y\,\frac{1}{8\pi\sqrt z}+\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\frac{e^{-\sqrt z\,|\tilde y-y|}}{8\pi\sqrt z} =\d\left(\zeta_y\,\frac{\sqrt z}{4\pi}-\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\G_z^{\tilde yy}\,\right) \ , } $$ where $\G_z^{\tilde yy}:=\G_z(\tilde y-y)$, $\tilde y\not=y$. Defining $$ \tilde\G_z:\C^n\to\C^n\qquad(\tilde \G_z\zeta)_y:=\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\G_z^{\tilde yy}\ , $$ one can take as $\Gamma_\Theta(z)$ the linear operator $$ \Gamma_\Theta(z)=\Theta+\frac{\sqrt z}{4\pi}-\tilde\G_z\ , $$ where $\Theta$ is any Hermitean $n\times n$ matrix.\par Hypotheses (h1) is satisfied with at least $\C\backslash\RE\subseteq Z_\Theta$ since $\X$ is finite dimensional (see Remark 8) and hypotheses (h2) is satisfied since $\G_z^y\notin H^2(\RE^3)$ for any $y\in Y$. In conclusion on can define the self-adjoint operator $\Delta_{\Theta}^Y$ with resolvent given by $$ (-\Delta_{\Theta}^Y+z)^{-1}=(-\Delta+z)^{-1}+ \sum_{y,\tilde y\in Y}\left(\Theta+\frac{\sqrt z}{4\pi}-\tilde\G_z \right)^{-1}_{y\tilde y}\G_z^y\otimes\G_{z^*}^{\tilde y}\ . %\qquad z\in\C\backslash\RE\ . $$ This coincides with the operator constructed in [AGHH, \S II.1.1]. \v\v {\bf Example 4.} {\it Infinitely many point interactions in three dimensions.}\v We take $\H=L^2(\RE^3)$, $A=\Delta$, $D(A)=H^2(\RE^3)\subset C_b(\RE^3)$. Considering then an infinite and countable set $Y\subset\RE^3$ such that $$ \inf_{\Sb y\not=\tilde y\\y,\tilde y\in Y\endSb}\, |y-\tilde y|=d>0\ ,\eqno(16) $$ we take as the linear operator $\tau$ the linear map $\tau_Y\phi:=\{\phi(y)\}_{y\in Y}$. The hypothesis (16) ensures its surjectivity and (see [AGHH, page 172]) $$ \tau_Y:H^2(\RE^3)\to\ell_2(Y)\,,\qquad\tau_Y\in B(H^2(\RE^3),\ell_2(Y)) $$ Proceeding as in the previous example one has then $$ \breve G(z):\LD\to\ell_2(Y)\,,\qquad\breve G(z)\phi=\{\G_z*\phi\,(y)\}_{y\in Y}\ , $$ and $$ G(z):\ell_2(Y)\to\LD $$ is the unique bounded linear operator which, on the dense subspace $$\ell_0(Y):=\left\{\zeta\in\ell_2(Y)\ :\ \#\,\text{\rm supp($\zeta$)}<+\infty\right\}\, ,$$ is defined by $$ G(z)\zeta=\sum_{y\in Y}\zeta_y\G_z^y\ , $$ i.e. $$\forall\,\zeta\in\ell_2(Y),\qquad G(z)\zeta=\G_z*\tau'_Y(\zeta)\ ,$$ where $\tau'_Y(\zeta)\in H^{-2}(\RE^3)$ is the signed Radon measure defined by $\tau'_Y(\zeta)(\phi)=\langle\zeta^*,\tau_Y\phi\rangle$. Taking $\zeta\in\ell_0(Y)$ one then obtain, proceeding as in example 2, $$ \left(\breve G(z)\cdot G(z)\zeta\right)_y =\zeta_y\,\frac{1}{8\pi\sqrt z}+\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\frac{e^{-\sqrt z\,|\tilde y-y|}}{8\pi\sqrt z} =\d\left(\zeta_y\,\frac{\sqrt z}{4\pi}-\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\G_z^{\tilde yy}\,\right) \ . $$ Posing $$ \tilde\G_z:\ell_0(Y)\to\ell_2(Y)\,,\qquad (\tilde\G_z\zeta)_y:=\sum_{\tilde y\not=y}\zeta_{\tilde y}\,\G_z^{\tilde yy}\ , $$ the operator $$ \tilde \Gamma(z):=\frac{\sqrt z}{4\pi}-\tilde\G_z $$ satisfies (10) and (11). Therefore, by Lemma 3 (with $E$ the canonical basis of $\ell_2(Y)$ and $D(\tilde\Gamma)=\ell_0(Y)$), $\tilde\G_z$ is closable and, denoting its closure by the same symbol, one can use the representation $$ \Gamma_\Theta(z)=\Theta+\frac{\sqrt z}{4\pi}-\tilde\G_z\ , $$ where $\Theta$ is any symmetric operator on $\ell_2(Y)$.\par By proposition 1, (h1) is satisfied when $\Theta$ is $\Gamma(z)$--bounded (see [K, thm. 1.1. chap. IV]), whereas (h2) is equivalent to $\tau'_Y(\zeta)\notin L^2(\RE^3)$ for any $\zeta\not=0$, which is always true since the support of $\tau'_Y(\zeta)$ is the null set $Y$. So, by theorem 1, one can define the self-adjoint operator $\Delta_{\Theta}^Y$ with resolvent given by $$ (-\Delta_{\Theta}^Y+z)^{-1}=(-\Delta+z)^{-1}+ \sum_{y,\tilde y\in Y}\left(\Theta+\frac{\sqrt z}{4\pi}-\tilde\G_z \right)^{-1}_{y\tilde y}\G_z^y\otimes\G_{z^*}^{\tilde y} %\qquad |z|\ge r_\Theta \ . $$ This coincides with the operator constructed (by an approximation method) in [AGHH, \S III.1.1]. \v\v {\bf Example 5.} {\it Singular perturbations of the Laplacian in $L^2(\RE^n)$, $n=3$, $n=4$, supported by regular curves.}\v We take $\H=L^2(\RE^n)$, $A=\Delta$, $D(A)=H^2(\RE^n)$, $n=3$ or $n=4$. Consider then a $C^2$ curve $\gamma:I\subseteq\RE\to\RE^n$ such that $C:=\gamma(I)$ is a one-dimensional embedded submanifold $C\subset\RE^n$ which, when unbounded, is, outside some compact set, globally diffeomorphic to a straight line (these hypotheses on $\gamma$ will be weakened in the next example). We will suppose $C$ to be parametrized in such a way that $|\dot\gamma|=1$.\par We take as linear operator $\tau$ the unique linear map $$ \tau_\gamma:H^2(\RE^n)\to L^2(I)\,,\qquad\tau_\gamma\in B(H^2(\RE^n),L^2(I)) $$ such that $$ \forall\phi\in C^\infty_0(\RE^n)\,,\qquad\tau_\gamma\phi(s):=\phi(\gamma(s))\ . $$ The existence of such a map is given by combining the results in [BIN, \S10] (straight line) with the ones in [BIN, \S 24] (compact manifold). By [BIN, \S 25] we have that $$\text{\rm Range}\,(\tau_\gamma)=H^s(I),\quad \tau_\gamma\in B(H^2(\RE^d),H^s(I)),\qquad s=2-\,\frac{n-1}{2} $$ and so we could take $\X=H^s(I)$. However we prefer to work with $\X=L^2(I)$ even if with this choice $\tau_\gamma$ is not surjective (but has a dense range). \smallskip\p 1) The case $n=3$.\p One has, proceeding similarly to examples 3 and 4, $$ \breve G(z):\ld3\to L^2(I)\,,\qquad\breve G(z)\phi =\tau_\gamma(\G_z*\phi) $$ and $$ G(z):L^2(I)\to\ld3\,,\qquad G(z)f=\G_{z}*\tau'_\gamma(f)\ , $$ where $\tau'_\gamma(\zeta)\in H^{-2}(\RE^3)$ is the signed Radon measure defined by $\tau'_\gamma(f)(\phi)=\langle f^*,\tau_\gamma\phi\rangle$. By Fourier transform one has equivalently $$ \F\cdot G(z)f(k)=\frac{1}{(2\pi)^{\frac{3}{2}}}\ \frac{1}{|k|^2+z}\, \int_I ds\,f(s)\,e^{-ik\cdot\gamma(s)}\,,\qquad f\in L^1(I)\cap L^2(I) \ , $$ so that, for any $f_1,f_2\in L^1(I)\cap L^2(I)$ one obtains $$ \eqalign{ &(z-w)\langle f_1,\breve G(w)\cdot G(z)f_2\rangle =(z-w)\,\langle G(w^*)f_1, G(z)f_2\rangle\cr =&\frac{(z-w)}{(2\pi)^3}\,\int_{I^2} dt\,ds\,{f^*_1(t)\,f_2(s)} \int_{\RE^3} dk\, \frac{e^{-ik\cdot(\gamma(t)-\gamma(s))}}{(|k|^2+w)\,(|k|^2+z)}\cr =&\frac{(z-w)}{2\pi^2}\,\int_{I^2} dt\,ds \,\frac{f_1^*(t)\,f_2(s)}{|\gamma(t)-\gamma(s)|} \int_0^\infty dr\, \frac{r\sin(r|\gamma(t)-\gamma(s)|)}{(r^2+w)\,(r^2+z)}\cr =&\int_{I^2} dt\,ds\,f_1^*(t)\,f_2(s) \ \frac{e^{-\sqrt w\,|\gamma(t)-\gamma(s)|}-e^{-\sqrt z\,|\gamma(t)-\gamma(s)|}}{4\pi\,|\gamma(t)-\gamma(s)|} \ .}\eqno(17) $$ Suppose now that, in the case $I$ is not compact, $$ \exists\,\lambda_0>0\ :\ \forall\,\lambda\ge \lambda_0,\qquad \sup_{t\in I}\int_{I}ds\, e^{-\lambda|\gamma(t)-\gamma(s)|}<+\infty\ .\eqno(18) $$ By (17) one can then define a linear operator ${\breve\Gamma_\epsilon}(z):L_0^2(I)\to L^2(I)$, $\epsilon>0$, satisfying (5) and (7), by $$ \breve\Gamma_\epsilon(z)f(t):= \int_{I} ds\,f(s)\left( \,\frac{\chi_\epsilon(t,s)}{4\pi\,|t-s|}-\frac{e^{-\sqrt z\,|\gamma(t)-\gamma(s)|}}{4\pi\,|\gamma(t)-\gamma(s)|}\,\right)\ , $$ where $\chi_\epsilon(t,s):=\chi_{[0,\epsilon]}(|t-s|)$ and $L^2_0(I):=\left\{f\in L^2(I)\, :\, \text{\rm $f$ has compact support}\right\}$. When $f\in C_0^1(I)$ one can then re-write $\breve\Gamma_\epsilon(z)f$ as $$\eqalign{ \breve\Gamma_\epsilon(z)f(t)=&\int_{I} ds\,(\,f(t)-f(s)\,) \,\G_z(\gamma(t)-\gamma(s))\cr +&f(t)\int_{I} ds \,\frac{\chi_\epsilon(t,s)}{4\pi\,|t-s|}-\frac{e^{-\sqrt z\,|\gamma(t)-\gamma(s)|}}{4\pi\,|\gamma(t)-\gamma(s)|}\cr -&\int_{I} ds\,\chi_\epsilon(t,s) \,\frac{f(t)-f(s)}{4\pi\,|t-s|} \ . } $$ The second term has, as a function of the parameter $\epsilon>0$, a derivative given by $(2\pi\epsilon)^{-1}f(t)$, and the last term is $z$--independent. Therefore the operator $\tilde\Gamma(z):C_0^1(I)\to L^2(I)$, $$\eqalign{ \tilde\Gamma(z)f(t):=& \int_{I} ds\,(\,f(t)-f(s)\,) \,\G_z(\gamma(t)-\gamma(s))\cr +f(t)\left(\,\frac{1}{2\pi}\,\log\left(\epsilon^{-1}\right)+\right.&\left.\int_{I} ds \,\frac{\chi_\epsilon(t,s)}{4\pi\,|t-s|}-\frac{e^{-\sqrt z\,|\gamma(t)-\gamma(s)|}}{4\pi\,|\gamma(t)-\gamma(s)|} \,\right)}\eqno(19) $$ is $\epsilon$--independent and satisfies (10) and (11) with $E=L^2(I)$ and $D(\tilde\Gamma)=C_0^1(I)$. Moreover, by Lemma 3, it is closable and its closure $\Gamma(z)$ satisfies (6) and (7).\par By Proposition 1, $\Gamma_\Theta(z)$ verifies hypothesis (h1) with $\Theta$ any symmetric operator on $L^2(I)$ which is $\Gamma(z)$--bounded (see [K, thm. 1.1. chap. IV]), whereas (h2) is satisfied since $\tau'_\gamma (f)\notin L^2(\RE^3)$ for any $f\not=0$, being the support of $\tau'_\gamma (f)$ given by the null set $C$. \par The corresponding self-adjoint family given by Theorem 1 has resolvents $$ (-\Delta_\Theta^\gamma+z)^{-1}\phi=(-\Delta+z)^{-1}\phi +\G_z*\tau'_\gamma\left(\,(\,\Theta+ \Gamma(z)\,)^{-1}\cdot\tau_\gamma(\G_z*\phi)\,\right)\ . $$ These give singular perturbations of the Laplacian of the same kind obtained (by a quadratic form approach) in [T]. \smallskip\p 2) The case $n=4$.\p Proceeding as in the case $n=3$ one obtains $$ \breve G(z):\ld4\to L^2(I)\,,\qquad\breve G(z)\phi =\tau_\gamma(\G_z*\phi)\ , $$ $$ G(z)f=\K_{z}*\tau'_\gamma(f),\quad \F\K_z(k):=\frac{1}{|k|^2+z},\quad k\in\RE^4 $$ and, for any $f_1,f_2\in L^1(I)\cap L^2(I)$, $$ \eqalign{ &(z-w)\langle f_1,\breve G(w)\cdot G(z)f_2\rangle =(z-w)\,\langle G(w^*)f_1, G(z)f_2\rangle\cr =&\int_{I^2} dt\,ds\,f_1^*(t)\,f_2(s) \,\left(\,\K_w(\gamma(t)-\gamma(s))-\K_z(\gamma(t)-\gamma(s))\,\right) \ .} $$ Since $$\eqalign{ |\K_z(x)|=&\frac{1}{4\pi^2|x|^2}\,(\,1+o\left(|x|\right)\,)\,,\qquad |x|\ll 1\ ,\cr |\K_z(x)|=&\frac{1}{2\,(2\pi)^{3/2}|x|^{3/2}}\,e^{-2\text{\rm Re}\sqrt z\,|x|}\, (\,1+o\left(1/|x|\right)\,)\,,\qquad |x|\gg 1\ ,} $$ when $\gamma$ satisfies (18) the linear operator $\breve\Gamma_\epsilon(z):L_0^2(I)\to L^2(I)$ $$ \breve\Gamma_\epsilon(z)f(t)= \int_{I} ds\,f(s) \,\left(\,\frac{\chi_\epsilon(t,s)}{4\pi^2\,|t-s|^2}-\K_z(\gamma(t)-\gamma(s))\,\right) $$ is well defined and satisfies (10) and (11) with $E=L^2(I)$ and $D(\breve\Gamma)=L_0^2(I)$. \par In four dimensions, due to the stronger (w.r.t. $\G_z$) singularity at the origin of $\K_z$, it is no more possible to perform the calculations leading to the analogue of the operator $\tilde \Gamma(z)$, and one is forced to use sesquilinear forms and to try then to apply Lemma 4. Defining for brevity $$ k_{(\epsilon)}(t,s):=\frac{\chi_\epsilon(t,s)}{4\pi^2\,|t-s|^2}\,,\qquad k_z(t,s):=\K_z(\gamma(t)-\gamma(s))\ , $$ one can re-write $\langle f_1,\breve\Gamma_\epsilon(z)f_2\rangle$, when $f_1,f_2\in C_0^1(I)$, as $$\eqalign{ &\langle f_1,\breve\Gamma_\epsilon(z)f_2\rangle=\int_{I^2}dt\, ds\,f_1^*(t) f_2(s)\,(\,k_{(\epsilon)}(t,s)-k_z(t,s)\,)\cr =&\int_{I^2}dt\, ds\,(\,f_1^*(t) f_2(s)-f_1^*(t) f_2(t)+f_1^*(s) f_2(s)\,)\,(\,k_{(\epsilon)}(t,s)-k_z(t,s)\,)\cr =&\frac{1}{2}\,\int_{I^2}dt\, ds\,(\,f_1^*(t)- f^*_1(s)\,)\,(\,f_2(t)-f_2(s)\,)\,(\,k_z(t,s)-k_{(\epsilon)}(t,s)\,)\cr &+\int_{I^2}dt\, ds\,f_1^*(t)f_2(t)\,(\,k_{(\epsilon)}(t,s)-k_z(t,s)\,)\cr =&\frac{1}{2}\,\int_{I^2}dt\, ds\,(\,f_1^*(t)- f^*_1(s)\,)\,(\,f_2(t)-f_2(s)\,)\,k_z(t,s)\cr &+\int_{I^2}dt\, ds\,f_1^*(t)f_2(t)\,(k_{(\epsilon)}(t,s)-k_z(t,s))\cr &-\frac{1}{2}\,\int_{I^2}dt\, ds\,(\,f_1^*(t)- f^*_1(s)\,)\,(\,f_2(t)-f_2(s)\,)\,k_{(\epsilon)}(t,s) \ . } $$ Similarly to the three dimensional case the second term has, as a function of the parameter $\epsilon>0$, a derivative given by $(2\pi^2\epsilon^2)^{-1}\smallint_I dt\, f^*_1(t)f_2(t)$, and the last term is $z$--independent. Therefore the sesquilinear form $$\tilde\E(z):C_0^1(I)\times C_0^1(I) \to\C\ ,$$ $$\eqalign{ \tilde\E(z)(f_1,f_2):= \frac{1}{2}\,&\int_{I^2}dt\, ds\,(\,f_1^*(t)- f^*_1(s)\,)\,(\,f_2(t)-f_2(s)\,)\,k_z(t,s)\cr +&\int_{I}dt\,f_1^*(t)f_2(t)\,\left(\,\frac{1}{2\pi^2\epsilon}+ \int_I ds\,(k_{(\epsilon)}(t,s)-k_z(t,s))\,\right) }\eqno(20) $$ is $\epsilon$--independent and satisfies (13) and (14). It is straightforward to check its closability (see the proof of proposition 2 in [T] if you get stuck), whereas (15) is a conseguence of (18). Being (h2) verified by the same argument as in the case $n=3$, by Lemma 4, Proposition 1 and Theorem 1, one has a self-adjoint family of self-adjoint operators with resolvents $$ (-\Delta_\Theta^\gamma+z)^{-1}\phi=(-\Delta+z)^{-1}\phi +\K_z*\tau'_\gamma\left(\,(\Theta+\Gamma(z)\,)^{-1}\cdot\tau_\gamma(\K_z*\phi)\,\right)\ , $$ where $\Gamma(z)$ is the operator corresponding to the closure of $\tilde\E(z)$ and $\Theta$ is any symmetric operator on $L^2(I)$ which is $\Gamma(z)$--bounded. This gives singular perturbations of the Laplacian of the same kind obtained in [Sh]. \par \v\v {\bf Example 6.} {\it Singular perturbations in $L^2(\RE^n)$ given by $d$--measures.}\v A Borel set $F\subset\RE^n$ is called a $d$--set, $d\in(0,n]$, if (see [JW2, Chap. II]) there exists a Borel measure $\mu$ in $\RE^n$ such that supp$\,(\mu)=F$ and $$ \exists\, c_1,\,c_2>0\ :\ \forall\, x\in F,\ \forall\,r\in(0,1),\qquad c_1r^d\le\mu(B_r(x)\cap F)\le c_2r^d\ ,\eqno(21) $$ where $B_r(x)$ is the ball of radius $r$ centered at the point $x$. By [JW2, chap. II, thm. 1], once $F$ is a $d$--set, $\mu_F$, the $d$--dimensional Haurdorff measure restricted to $F$, always satisfies (21) and so $F$ has Hausdorff dimension $d$ in the neighbourhood of any of its points. From the definition there also follows that a finite union of $d$--sets which intersect on a set of zero $d$--dimensional Hausdorff measure is a $d$--set. Examples of $d$--sets are $d$--dimensional Lipschitz manifolds (use examples 2.1 and 2.4 in [JW1]) and (when $d$ is not an integer) self-similar fractals of Hausdorff dimension $d$ (see [JW2, chap. II, example 2], [Tr, thm. 4.7]). \par Denoting by $j_F:F\to\RE^n$ the restriction to the $d$--set $F$ of the identity map, we take as the linear operator $\tau$ the unique continuous map ($00\ :\ \forall\, x\in \RE^d,\ \forall\,r\in(0,1],\qquad \mu(B_r(x))\le cr^d\ . $$ Then, by [JW, lemma 1, chap. VIII], when $$p=\frac{2d}{n-2s_*},\qquad 02)$ point interacting particles (see [MF], [DFT] and references therein). \v\v {\bf Example 7.} {\it Singular perturbations of the d'Alembertian in $L^2(\RE^4)$ supported by time-like straight lines.}\v We take $\H=L^2(\RE^4)$, $$A=\square:=-\Delta_{(1)}\otimes\uno+\uno\otimes\Delta_{(3)}\,,$$ $\Delta_{(d)}$ being the Laplacian in $d$ dimensions, and ($h\in\RE$, $k\in\RE^3$ denoting the variables dual to $t\in\RE$, $x\in\RE^3$) $$D(\square)=\left\{\Phi\in L^2(\RE^4)\ :\ (h^2-|k|^2)\F\Phi(h,k)\in L^2(\RE^4)\right\}\ .$$ Let $l(s)=y+ws$, $y,w\in\RE^4$, be a time-like straight line, i.e. $$ w=(\gamma_v,\gamma_v\,v),\quad v\in\RE^3,\quad |v|<1,\quad \gamma_v:= \left(1-|v|^2\right)^{-1/2}\ . $$ Consider now the unique surjective linear operator $$ \tau_0: D(\square)\to H^{-\frac{1}{2}}(\RE)\,,\qquad\tau_0\in B(D(\square), H^{-\frac{1}{2}}(\RE)) $$ such that $$ \forall\Phi\in C^\infty_0(\RE^4),\qquad\tau_0\Phi(s):=\phi(s,0)\ . $$ The existence of such a $\tau_0$ is ensured by [H, thm. 2.2.8] (see the next example). Let then $\Pi_{y,v}$ be the unitary operator which compose any function in $L^2(\RE^4)$ with the Lorentz boost corresponding to $v$ and then with the translation by $y$, so that $\Pi_{y,v}\in B(D(\square),D(\square))$. Defining $$ \tau_{y,v}:=\tau_0\cdot\Pi_{y,v}:D(\square)\to H^{-\frac{1}{2}}(\RE)\,,\qquad\tau_{y,v}\in B(D(\square), H^{-\frac{1}{2}}(\RE)) $$ one has $$ \forall\Phi\in C^\infty_0(\RE^4),\qquad\tau_{y,v}\phi(s):=\phi(l(s))\ . $$ %Let us remark that nobody (to the author knownledge) has studied the %map $\tau$ when the straight line $\gamma$ is replaced by an %arbitrary regular time-like path. \p We begin studing the self-adjoint extensions given by $\tau_0$. By Fourier transform (here and below $z\in\C\backslash\RE$) one obviously has $$ \F\cdot(\square+z)^{-1}\Phi(h,k)=\frac{\F\Phi(h,k)}{-h^2+|k|^2+z}\ . $$ So, since, as $R\uparrow\infty$, $$ \frac{1}{\sqrt {2\pi}}\int_0^\infty dh\,\frac{1}{\left|-h^2+R^2+z\right|^2}\ \sim\frac{c}{4R^3}\ , $$ by H\"older inequality and Riemann--Lebesgue lemma there follows that $$ \forall\,\Phi=\phi\otimes\varphi\in L^2(\RE)\otimes H^s(\RE^3),\ s>0,\qquad (\square+z)^{-1}\Phi\in C_b^0(\RE^4) $$ and, by Fubini theorem, $$ \eqalign{ &[(-\square+z)^{-1}\Phi]\,(t,x)\cr =&{1\over{(2\pi)^2}}\,\int_{\RE^4}dh\,dk\,e^{iht}e^{ik\cdot x}{{\F\Phi (h,k)}\over{-h^2+|k|^2+z}}\cr =&{1\over{(2\pi)^{1/2}}}\,\int_{\RE}dh\,e^{iht}\F\phi(h) \left(\left(-\Delta-h^2+z\right)^{-1}\varphi\right)(x)\cr =&{1\over{(2\pi)^{1/2}}}\,\int_{\RE}dh\,e^{iht}\F\phi(h) \int_{\RE^3}dy\,\vp(y)\,\frac{e^{-\sqrt{-h^2+z}\,|x-y|}}{4\pi\,|x-y|}\cr =& \int_{\RE^3}dy\,\frac{\vp(y)}{4\pi\,|x-y|}\, \left(e^{-|x-y|\,\sqrt{\Delta_{(1)}+z}}\,\phi\right)(t) \ . } $$ Here $\text{\rm Re}\sqrt{-h^2+z}>0$; this choice will be always assumed in the sequel without further specification. The above calculation then gives $$ \gather \breve G(z):L^2(\RE)\otimes H^{s}(\RE^3)\to C_b^0({\RE})\, ,\quad s>0,\\ \left(\breve G(z)\phi\otimes\varphi\right)\,(t):= \int_{\RE^3}dy\,\frac{\varphi(y)}{4\pi\,|y|}\, \left[e^{-|y|\,\sqrt{\Delta_{(1)}+z}}\phi\right](t) \endgather $$ and $$ \gather G(z):H^{1/2}(\RE)\to L^2(\RE^4)\\ (G(z)\phi)\,(t,x):=\frac{1}{4\pi\,|x|}\, \left[e^{-|x|\,\sqrt{\Delta_{(1)}+z}}\phi\right](t)\ . \endgather $$ Let us note that $\breve G(z)$ extends to a continuous linear operator from $L^2(\RE^4)$ to $H^{-1/2}(\RE)$ since $$ \eqalign{ &\|\breve G(z)\phi\otimes\varphi\|^2_{H^{-1/2}}\cr \le&\, \int_\RE dt\left(\,\int_{\RE^3}dy\,\frac{|\varphi(y)|}{4\pi\,|y|}\, \left|\left[(-\Delta_{(1)}+1)^{-1/4}\cdot e^{-|y|\,\sqrt{\Delta_{(1)}+z}} \phi\right](t)\right|\,\right)^2\cr \le&\,\|\varphi\|_{L^2}^2 \,\left\langle\left(-\Delta_{(1)}+1\right)^{-1/2}\cdot \int_0^\infty dR\, e^{-2R\,\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)}\phi,\phi\right\rangle\cr =&\,\frac{1}{2}\,\|\varphi\|_{L^2}^2 \,\left\langle\left(-\Delta_{(1)}+1\right)^{-1/2}\cdot \left(\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)\right)^{-1}\phi,\phi \right\rangle\cr \le&\,\frac{1}{2}\,\|\phi\otimes\varphi\|_{L^2}^2 \,\left\|\left(\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)\right)^{-1/2}\right\|_{L^2,H^{-1/2}}\ . } $$ Similarly $G(z)$ is a continuous linear operator from $H^{1/2}(\RE)$ to $L^2(\RE^4)$ since $$ \eqalign{ &\|G(z)\phi\|^2_{L^2}\cr =&\int_0^\infty dR\,\left\| e^{-R\,\sqrt{\Delta_{(1)}+z}} \phi\right\|^2_{L^2}\cr =&\,\left\langle\int_0^\infty dR\, e^{-2R\,\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)}\phi,\phi \right\rangle\cr =&\,\frac{1}{2} \,\left\langle \left(\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)\right)^{-1}\phi,\phi \right\rangle\cr \le&\,\frac{1}{2}\,\|\phi\|_{H^{1/2}}^2 \,\left\|\left(\text{\rm Re}\left(\sqrt{\Delta_{(1)}+z}\,\right)\right)^{-1/2}\right\|_{H^{1/2},L^2}\ . } $$ We now look for the map $\Gamma(z)$. Since $$ \eqalign{ &(z-w)\,\F\cdot\breve G(w)\cdot G(z)\phi(h)\cr =& \frac{z-w}{(2\pi)^3}\,\F\phi(h)\int_{\RE^3} dk\, \frac{1}{(-h^2+|k|^2+w)(-h^2+|k|^2+z)}\cr &=\frac{z-w}{2\pi^2}\,\F\phi(h)\int_0^\infty dr\, \frac{r^2}{(-h^2+r^2+w)(-h^2+r^2+z)}\cr =&\frac{1}{4\pi}\,\left(\sqrt{-h^2+z}-\sqrt{-h^2+w}\,\right)\F\phi(h) } $$ one defines $$ \Gamma(z):=\frac{1}{4\pi}\,\sqrt{\Delta_{(1)}+z}\ :H^{1/2}(\RE)\to H^{-1/2}(\RE)\ . $$ Of course we can view $\Gamma(z)$ as a (unbounded) closed and densely defined linear operator on the Hilbert space $H^{-1/2}(\RE)$. Therefore, by Proposition 1, $\Gamma_\Theta(z)$ satisfies (h1) (with $Z_\Theta=\rho(\square)$) for any symmetric $\Theta:H^{1/2}(\RE)\to H^{-1/2}(\RE)$ which is $\Gamma(z)$--bounded. It is immediate, by Fourier transform, to check the validity of (h2). Therefore the trace $\tau_0$ gives rise to the family of self-adjoint extensions $\square^0_\Theta$ with resolvent $$ (-\square^0_\Theta+z)^{-1}=(-\square+z)^{-1} +G(z)\cdot\left(\,\Theta+\frac{1}{4\pi}\,\sqrt{\Delta_{(1)}+z}\,\right)^{-1} \cdot\breve G(z)\ . $$ By our definition of $\tau_{y,v}$ we have, since $\Pi_{y,v}$ commutes with $\square$, $$ \breve G_{y,v}(z):=\tau_{y,v}\cdot R^0(z)=\breve G(z)\cdot\Pi_{y,v} $$ and $$ C_{L^2}^{-1}\cdot\breve G_{y,v}(z^*)'=\Pi^*_{y,v}\cdot G(z)\ . $$ This immediately implies that one can use the same $\Gamma_\Theta(z)$ as before and so the trace $\tau_{y,v}$ gives rise to the family of self-adjoint extensions $\square^{y,v}_\Theta$ with resolvent $$ (-\square^{y,v}_\Theta+z)^{-1}=(-\square+z)^{-1} +\Pi^*_{y,v}\cdot G(z)\cdot\left(\,\Theta+\frac{1}{4\pi}\,\sqrt{\Delta_{(1)}+z}\,\right)^{-1} \cdot\breve G(z)\cdot\Pi_{y,v}\ , $$ so that the following kind of Poincar\'e--invariance holds: $$ D(\square^{y,v}_\Theta)=\Pi^*_{y,v}(D(\square^0_{\Theta}))\ \quad \text{\rm and}\quad\ \square^{y,v}_\Theta=\Pi^*_{y,v}\cdot\square^0_{\Theta}\cdot \Pi_{y,v}\ . $$ \v\v {\bf Example 8.} {\it Singular perturbation given by traces on the H\"ormander spaces $H_{\varphi}(\RE^n)$. }\v Given any positive functions $\varphi$ on $\RE^n$, $\varphi\in\K$ will mean $$ \exists\, c,\,N>0\ :\ \forall\,x,\,y\in\RE^n,\qquad \varphi(x+y)\le (\,1+c\,|x|\,)^N\varphi(y)\ . $$ Then we define, following [H, \S II.2.2] (also see [VP]), the Hilbert space $H_\varphi(\RE^n)$, $\varphi\in\K$, as the set of tempered distribution $f$ such that $\F f$ is a functions and $$ \|f\|^2_\varphi:=\int_{\RE^n}|\varphi(k)\F f(k)|^2\, dk<+\infty\ . $$ By [H, example 2, \S II.2.2], for any (complex--valued) polynomial $p$ one has $\tilde p\in\K$, where $$ \tilde p(x):=\left(\,\sum_{|\alpha|\ge 0}\left|\,D^\alpha\,p(x)\right|^2\,\right)^{1/2} $$ and $D^\alpha:=\partial^{|\alpha|}/\partial_1^{\alpha_1} \cdots\partial_n^{\alpha_n}$. Moreover, by [H, thm. 2.1.1], $$\vp_1+\vp_2,\ \vp_1\vp_2,\ \vp_1\wedge\vp_2,\ \vp_1\vee\vp_2,\ \vp^s,\ \mu*\vp\,\in\K$$ for any $\vp,\,\vp_1,\,\vp_2\,\in\K$, for any $s\in\RE$ and for any positive measure $\mu$ such that $\mu*\vp(x)\not=\infty$ for some (and hence any) $x\in\RE^n$.\par In connection with the previous examples note that $$\varphi(x)=(1+|x|^2)^{s/2},\ s\in\RE\quad\Longrightarrow\quad H_\varphi(\RE^n)=H^s(\RE^n)$$ and $$ \varphi(t,x)=(1+(-t^2+|x|^2)^2)^{1/2},\ t\in\RE,\ x\in\RE^3 \quad\Longrightarrow\quad H_\varphi(\RE^4)=D(\square)\ . $$ The dual space of $H_\vp$ can be explicitely characterized (see [H, thm. 2.2.9]) as $$ H'_\vp\simeq H_{1/\vp}\ . $$ As regards the relation between different spaces, by [H, thm. 2.2.2] one has $$ \vp_1\le c\,\vp_2,\quad\vp_1,\vp_2\in\K\quad\Longrightarrow\quad H_{\varphi_1}\subseteq H_{\varphi_2}\ , $$ the embedding being continuous. Therefore, for any $\vp\in\K$ such that $\vp\ge c>0$, one has $H_\vp\subseteq L^2(\RE^n)$; $H_\vp$ is then dense in $L^2(\RE^n)$ since $C^\infty_0(\RE^n)$ is dense in $H_\vp(\RE^n)$ for any $\vp\in \K$ (see [H, thm. 2.2.1]). The regularity of elements in $H_\vp(\RE^n)$ is given by [H, thm. 2.2.7]: $$ (1+|x|)^k/\vp(x)\in L^2(\RE^n)\quad\Longrightarrow\quad H_\vp(\RE^n)\subset C^k(\RE^n)\ , $$ the embedding being continuous. \par Let us now come to the trace operator on $H_\vp(\RE^n)$ (see [H, thm. 2.2.8], [VP, \S 6]). We write $\RE^n=\RE^d\oplus\RE^{n-d}$, $1\le d\le n-1$, $x=(\tilde x,\hat x)$, $\tilde x\in\RE^d$, $\hat x\in\RE^{n-d}$. Suppose that $$ \left(\int_{\RE^{n-d}} \frac{1}{\vp^2(0,\hat x)}\,d\hat x\right)^{-1/2}\,<+\infty\ . $$ Then there exists an unique surjective linear operator $\tau_{(d)}$ $$ \tau_{(d)}:H_\vp(\RE^n)\to H_{\tilde\vp}(\RE^d)\,,\qquad \tau_{(d)}\in B(H_\vp(\RE^n),H_{\tilde\vp}(\RE^d))\ , $$ $$ \tilde\vp(\tilde x):=\left(\int_{\RE^{n-d}} \frac{1}{\vp^2(\tilde x,\hat x)}\,d\hat x\right)^{-1/2}\ , $$ such that $$ \forall\,\phi\in C_0^\infty(\RE^n)\,,\qquad\tau_{(d)}(\phi)(\tilde x)=\phi(\tilde x,0)\ . $$ The reader can check that the case $\varphi(t,x)=(1+(-t^2+|x|^2)^2)^{1/2}$, $d=1$, reproduces the trace $\tau_0$ given in the previous example. \par The trace $\tau_{(d)}$ can be generalized to cover the case of non-linear subsets in the following way: let $\mu\in H'_\phi(\RE^n)$ (for example $\mu$ could be the Hausdorff measure of some subset of $\RE^n$ but more general distributions are allowed) for which there exists $\tilde\phi\in\K$ such that $$ \int_{\RE^n}\frac{\phi^2(x-y)}{\varphi^2(x)\,\tilde\phi^2(y)}\,dy0$) $$\psi(D):H_\vp(\RE^n)\to L^2(\RE^n)\,,\qquad \psi(D)\Phi:=\F^{-1}(\psi\,\F \Phi)\ , $$ where $\psi$ is a real-valued Borel function such that $$\frac{1}{c}\,\vp(x)\le 1+|\psi(x)|\le c\,\vp(x)\ .$$ By Fourier transform one has, if $\tau_\mu$ is defined as above, $$ G(z)f=G^\mu(z)f:=\frac{1}{(2\pi)^{n/2}}\,\F^{-1}\left(\frac{\F f*\F\mu}{-\psi+z^*}\right)\ . $$ Therefore (h2) is equivalent to $\F f*\F\mu\notin \ld{n}$, i.e. $f\mu\notin\ld{n}$. This condition is surely satisfied when the support of $\mu$ is a set of zero Lebesgue measure. \par By lemma 2 we have then, for any $f_1,f_2\in H_{\tilde\phi}(\RE^n)$, %(here we use the %reflexivity of $H_{\tilde\phi}(\RE^n)$ and identify %$H_{\tilde\phi}''(\RE^n)$ with $H_{\tilde\phi}(\RE^n)$) $$ \hat\Gamma(z)f_1(f_2)=f_2\mu((\tilde G^\mu-G^\mu(z))f_1)\ , \eqno(22) $$ where $$ \tilde G^\mu:=\frac{G^\mu(z_0)+G^\mu(z_0^*)}{2}\,,\qquad z_0\in\rho(\psi(D))\ . $$ In the case $$ \int_{\RE^n}\frac{\phi^2(x-y)}{\tilde\phi^2(x)\,\vp^2(y)}\,dy