\magnification=1200
\baselineskip=20pt
\def\epf{\square}
\def\ret{r^{\eta/4}}
\def\rme{r^{-\eta/4}}
\def\lto{L^1(a,\infty)}
\def\re{{\rm Re}\,}
\def\lor{L^1(\real)}
\def\pmin{\psi_{-}}
\def\ppl{\psi_{+}}
\def\hvp{ H_{Q,p}}
\def\lp{L^p(\real)}
\def\lo{L^1(\real)}
\def\mpg{M_p(\alpha,\gamma)}
\def\tgf{t^{\gamma/4}}
\def\tgmf{t^{-\gamma/4}}
\def\lto{L^1(a,\infty)}
\def\re{{\rm Re}\,}
\def\lor{L^1(\real)}
\def\pmin{\psi_{-}}
\def\ppl{\psi_{+}}
\def\real{{\bf R}}
\def\epf{{\bf //}}
\def\hvg{H_{Q,p}}
\def\mag{M(\alpha,\gamma)}
\def\plt{|\gamma|\leq 2\alpha/p}
\def\pgt{|\gamma|> 2\alpha/p}
\def\rmz{\real\setminus\{0\}}
\def\cont{{\rm C }}
\def\psiv{\Psi_+(x,V)}
\def\ppv{\psi_+(r,V)}
\def\pmv{\psi_-(r,V)}
\def\ppp{\psi_+'(r,V)}
\def\pmp{\psi_-'(r,V)}
\def\fhk{f(x+h, y+k)}
\def\fh{f(x+h,y)}
\def\fk {f(x,y+k)}
\def\fx{f(x,y)}
\def\ino{\int_0^1}
\def\ffi{\varphi}
\def\imi{\int_{-\infty}^\infty}
\def\test{C_0^\infty(\real)}
\def\uvt{ U_Q(t)}
\def\uzv{U_0(t)}
\def\upq{U_{Q,p}(t)}
\centerline{\bf PERTURBATIONS OF THE WIGNER-VON NEUMANN POTENTIAL}
\medskip
\centerline{\bf LEAVING THE EMBEDDED EIGENVALUE FIXED}
\medskip
\vskip.5cm
\centerline{ \bf J. Cruz-Sampedro\footnote*{
Research supported in part by CONACYT, 32146-E, Mexico.}}
\medskip
\centerline{Instituto de Ciencias B\'asicas e Ingenier\'\i a, UAEH}
\centerline{sampedro@uaeh.reduaeh.mx}
\medskip
\centerline{ \bf I. Herbst\footnote{**}{
Research partially supported by NSF grant DMS-96000056.}}
\medskip
\centerline{Mathematics Department, University of Virginia}
\centerline{iwh@weyl.math.virginia.edu}
\medskip
\centerline{\bf and}
\medskip
\centerline{ \bf R. Mart\'\i nez-Avenda\~no}
\centerline{ Department of Mathematics, Michigan State University}
\centerline{ruben@math.msu.edu}
\vskip1cm
{\bf Abstract.}
We investigate the Schr\"odinger operator $H=-d^2/dx^2+(\gamma/x)\sin
\alpha x+V$, acting in $ L^p(\real)$, $1\leq p<\infty$, where $\gamma \in \real
\setminus
\{ 0 \} $, $\alpha >0$, and $V \in L^1(\real)$. For
$\plt $ we show that $H$ does not have
positive eigenvalues. For $\pgt$ we show that the set of
functions $V\in L^1(\real)$,
such that $H$ has a positive eigenvalue embedded in the essential
spectrum $\sigma_{\rm ess}(H)=[0,\infty)$, is a smooth unbounded
sub-manifold
of $L^1(\real)$ of codimension one.
\vfill\eject
\centerline{\bf Perturbations du Potentiel Wigner-Von Neumann}
\medskip
\centerline{\bf Qui Fixent la Valeur Caract\'eristique Immerg\'ee}
\vskip1cm
{\bf R\'esum\'e.} On examine l'op\'erateur de Schr\"odinger
$H=-d^2/dx^2+(\gamma/x)\sin
\alpha x+V$ d\'efini dans $ L^p(\real)$, $1\leq
p<\infty$,
o\`u $\gamma \in \real \setminus
\{ 0 \} $, $\alpha >0$, et $V \in L^1(\real)$. Si
$|\gamma|\leq 2\alpha/p$, on montre que $H$ n'a aucune valeur caract\'eristique
positive.
Si $|\gamma|> 2\alpha/p$, on montre que l'ensemble des fonctions
$V\in L^1(\real)$, telles que
$H$ a une valeur caract\'eristique positive immerg\'ee dans
le spectre essentiel $\sigma_{\rm ess}(H)=[0,\infty)$,
est une sous-vari\'et\'e lisse non-born\'ee de $L^1(\real)$ de
codimension \'egale \`a un.
\vfill\eject
\noindent{\bf 1. Introduction}
\medskip
In this paper we consider Schr\"odinger operators of the form
$$\hvp=-{d^2\over dx^2}+Q, \eqno(1.1)$$
acting in $\lp$, $1\leq p<\infty$, where
$Q=W+V$, $W(x)=(\gamma/x)\sin\alpha x$,
$\alpha>0$ and $\gamma\in \real\setminus\{0\}$ are
constants, and $V$ is a real-valued function in $\lo$.
To give a precise definition of the operator $\hvp$ we use the
{\it Feynman-Kac formula}. For $\displaystyle{f\in\cup_{p\geq 1}\lp}$
and $t\geq 0$ we define
$$ U_Q(t)f(x)=E_x\left(\exp\left\{-\int_0^tQ(b(s))ds\right\}f(b(t))
\right),\eqno(1.2) $$
where $E_x$ denotes the expectation
with respect to Brownian motion starting at $x$ with Brownian
transition function given by
$$p_t(x,y)={\exp\left({-(x-y)^2/ 4t}\right)\over\sqrt{4\pi t}},
\qquad\qquad x,y\in\real,\quad t\geq 0. \eqno(1.3) $$
We define $\hvp$ to be the negative of the
infinitesimal generator of
the C$_0$-semigroup $(\upq; t\geq 0) $, $ 1\leq p<\infty$,
defined for $f\in\lp$ and $t\geq 0$ by $\upq f=U_Q(t)f$.
Various classes of operators which contain the ones defined above
have been investigated in
[6, 12, 16, 17, 18, 26, 28, 30], and it is well known that
the spectrum $\sigma(\hvp)$ is $p$-independent and that
$\sigma_{\rm ess}(\hvp)=[0,\infty)$ for all $p\geq 1$.
\medskip
Schr\"odinger operators of the form (1.1) were introduced
by {\it Wigner} and {\it Von-Neumann} [29] in order to construct
an example of a Schr\"odinger operator, acting in $L^2(\real^3)$,
with a spherically symmetric potential which vanishes at
infinity and possesses a positive
eigenvalue embedded in the continuum. The
significance of the Wigner-Von Neumann example lies in
the fact that at the time it contradicted physical intuition,
which predicted that
bound states of positive energy could not occur if the potential
tended to zero at infinity.
\medskip
In this paper we study the structure of the set of functions
$V\in\lo$ for which the operator $\hvp$ has a positive eigenvalue.
\medskip
Our main result is:
\proclaim Theorem 1.1. Let $\hvp$ be as in (1.1).
If $\plt$, then $\hvp$ does not have positive eigenvalues.
If $\pgt$, then the set
of functions $V\in \lo$ such that $\hvp$ has a positive eigenvalue
embedded
in the essential spectrum $\sigma_{\rm ess}(\hvp)=[0,\infty)$ is
a smooth unbounded sub-manifold of $\lo$ of codimension one.
In addition, if $V$ belongs to this sub-manifold then $\alpha^2/4$
is the unique positive eigenvalue of $\hvp$.
It is well known that the
eigenvalues in the discrete spectrum are, in an appropriate setting,
stable under perturbations. On the other hand, it is also known that
embedded eigenvalues
in the continuum are rather unstable [1, 2, 9, 10, 20]. In [2]
{\it Agmon}, {\it Herbst}, and {\it Skibsted} prove that generically, in
a Baire category sense, arbitrarily small perturbations of a generalized
$N$-body Hamiltonian remove all non-threshold eigenvalues embedded in
the continuum, and conjecture that the set of perturbations that
preserve a non-threshold embedded eigenvalue is something like
a differentiable manifold. The result presented in this paper shows
that the above conjecture is true for
the simplest Schr\"odinger operators which possess
an eigenvalue embedded in the continuum. A similar
result for $p=2$ was announced in [11] without proof.
For $\alpha>0$ and $\gamma \in \rmz$, let $\mag$
be the set of functions $V\in\lo$ such
that, for some $k>0$, the differential equation
$$-\psi''+\gamma{\sin\alpha r\over r}\psi+V\psi=k^2\psi,
\qquad r\in \real,\eqno(1.4)$$
has a nonzero solution that goes to zero as $|r|$ goes to infinity.
We say that a function $\psi$ is a solution of
this differential equation
if it is continuously differentiable, $\psi'$ is absolutely continuous,
and (1.4) holds almost everywhere.
Local existence of solutions to (1.4) is
well known. We also prove
\proclaim Theorem 1.2. Let $\mag$ be as defined above. Then $\mag$ is
a smooth unbounded sub-manifold of $\lo$ of codimension one.
In addition, if $V\in\mag$ then $k=\alpha/2$.
\medskip
Using the terminology of [19],
$\mag$ is the set of functions $V$ in
$\lo$ such that $H_{Q,2}$ has a half-bound state of positive energy.
\medskip
To prove the results stated above we determine,
following {\it Cassell} [7], the exact asymptotic behavior at
infinity of the solutions to (1.4) and then identify the set of
functions $V$
in $\lo$ that produce positive eigenvalues of $\hvp$ with the zero
set of a smooth function on $\lo$ for which zero is a regular value.
\medskip
Schr\"odinger operators with eigenvalues
in the
continuous spectrum have also been investigated in [3, 14, 23], and the
asymptotic behavior of the solutions of (1.4) for various classes of
potentials
has also been studied in [4, 7, 13, 15, 22]. For
perturbations of embedded eigenvalues in situations which are
relevant to the
automorphic Laplacian and $N$-body Schr\"odinger operators
see [2, 5, 9, 10, 20, 25]. In a
different context, results of the type presented here
have been obtained in [21].
\medskip
This paper is organized as follows.
In Section 2 we investigate the asymptotic behavior at infinity of solutions
to (1.4) and establish the existence of solutions that vanish at
infinity. In Section 3 we prove the main results.
In the Appendix we establish the connection between the eigenfuctions
of $\hvp$ and the solutions of (1.4) that belong to $\lp$.
\medskip
{\bf Acknowledgments.} We thank S. Sontz for useful comments.
\medskip
\noindent{\bf 2. Existence of Solutions that Vanish at Infinity}
\vskip.3cm
In this section we follow {\it Cassell} [7] to determine
the asymptotic behavior as $r$ goes to infinity of the solutions to
(1.4). We will prove
\medskip
{\bf Theorem 2.1} {\sl For $\alpha>0$, $\gamma\in\rmz$, $k>0$,
and $V\in\lo$ we have:
\vskip.3cm
i) If $k\not =\alpha/2$, then (1.4) has solutions $\phi$
and $\psi$ such that, as $r$ goes to $+\infty$,
$$\phi(r)=\cos kr+o(1)\qquad \hbox{and}\qquad \psi(r)=\sin kr+o(1),$$
with
$$\phi'(r)=-k\sin kr+o(1)\qquad\hbox{and}\qquad\psi'(r)=k\cos kr+o(1).$$
ii) If $k=\alpha/2$, then (1.4) has solutions $\phi$ and $\psi$
such that, as $r$ goes to $ +\infty$,
$$\phi(r)=r^{-\gamma/{2\alpha}}(\cos kr+o(1))\qquad \hbox{and}\qquad
\psi(r)=r^{\gamma/{2\alpha}}(\sin kr+o(1)),$$
with
$$\phi'(r)=-kr^{-\gamma/2\alpha}(\sin kr+o(1))\qquad \hbox{and}\qquad
\psi'(r) =kr^{\gamma/2\alpha}(\cos kr+o(1)).$$}
\medskip
{\it Proof.} Setting $\xi(r)=\psi(r/k)$, $\sigma=\alpha/k>0$, and
$\eta=\gamma/k\in\rmz$ we
find that
$\xi$ satisfies
$$ -\xi'' + \eta {\sin\sigma r \over r} \xi + W \xi =
\xi,\eqno(2.1)$$
where $W(r)=V(r/k)/k^2\in L^1(\real)$.
Using the transformation
$$x=\pmatrix{\cos r&-\sin r\cr\sin r&\cos r\cr} \pmatrix{\xi\cr
\xi'},$$
we see that (2.1) is equivalent to
$$ x'= A(r) x,\eqno(2.2)$$
where
$$ A(r)=a(r) \pmatrix{\sin r\cos r&\sin^2 r\cr-\cos^2 r&-\sin r\cos
r\cr},$$
and $\displaystyle{a(r)= -\eta {\sin\sigma r \over r} - W(r)}$.
\medskip
Next we write $A(r)=(\eta/r)G(r) + R(r)$,
where $R(r)$ is the
$L^1$-matrix given by
$$R(r)=-W(r)\pmatrix{\sin r\cos r&\sin^2 r\cr-\cos^2 r&-\sin r\cos
r\cr},\eqno(2.3)$$ and
$$G(r)\equiv \pmatrix{g_1(r)&g_2(r)\cr g_3(r)&-g_1(r)\cr}, \eqno(2.4)$$
with
$$g_1(r)=-{1\over4}(\cos(\sigma-2)r-\cos(\sigma+2)r),$$
$$g_2(r)=-{1\over4}(2\sin\sigma r-\sin(\sigma+2)r-\sin(\sigma-2)r),$$
and
$$g_3(r)={1\over4}(2\sin\sigma r+\sin(\sigma+2)r+\sin(\sigma-2)r).$$
Now we decompose
$G$ as $G=G_1+G_2$, where
$G_1=0$ and $G_2=G$ for $\sigma\not=2$, and
$$G_1=\pmatrix{-{1\over4}&0\cr0&{1\over 4}\cr}$$
and
$$G_2(r)={1\over 4}\pmatrix{\cos 4r&-2\sin 2r+\sin 4r\cr
2\sin 2r+\sin 4r &-\cos 4r\cr}$$
for $\sigma=2$.
\medskip
Setting
$ S(r) = I + (\eta/r) G_2^* $, a crude approximation to a solution of
$S'=(\eta/r)G_2S$,
where
$$ G_2^*(r) \equiv \int_0^r G_2(u)du,$$
we find that if $a$ is large then $S(r)^{-1}$ exists
for $r\geq a$ and $\sup\{\|S(r)^{-1}\|:r>a\}<\infty$.
Hence setting
$$\tilde R=S^{-1}((\eta/r)^2(G G_2^* - G_2^* G_1) + R S +(\eta/r^2) G_2^*)$$
we have that $\tilde R\in\lto$ and defining
$B=(\eta/r)G_1+\tilde R$ we have that
$$S B = A S - S' .\eqno(2.5)$$
Therefore
setting $ x=S(r) y $ and using (2.5) we find that (2.2) is equivalent to
$$ y' =B(r) y. \eqno (2.6) $$
\medskip
To finish the proof we proceed as follows:
\medskip
i) If $\sigma\not=2$ then $B=\tilde R\in\lto$. Hence,
proceeding as in the proof of Theorem XI.65 of [22] we find that, as $r$
goes to $ +\infty$,
(2.6) has a fundamental matrix $X=I+o(1)$, where $I$ denotes the $2\times 2$
identity matrix. Thus (2.1) has solutions $\psi_1$, $\psi_2$ such that
$$\psi_1(r)=\cos r+o(1),\qquad \psi_2(r)=\sin r+o(1),$$
$$\psi_1'(r)=-\sin r+o(1),\qquad\hbox{and}\qquad \psi_2'(r)=\cos r+o(1),$$
from which i) of Theorem 2.1 follows.
\medskip
ii) If $\sigma=2$ and $\gamma >0$, then the change of
variables $\tau = \eta \log r$ transforms (2.6) into
$$ {d \ffi \over {d \tau}} = \left(G_1+
L \right) \ffi,\eqno(2.7)$$
where $L$ is in $L^1(\tau_0,\infty)$ for some $\tau_0$ independent of $V$.
It is easily verified that this last system of O.D.E.s satisfies
the conditions of a theorem due to Levinson. See Theorem 8.1 in Ch. 3 of
[8]. Thus, as $\tau$ goes to $ +\infty$, (2.7) has solutions $\ffi_1$ and
$\ffi_2$ such that
$$\lim_{\tau\to\infty}\exp(\tau/4)\ffi_1(\tau)=
\pmatrix{1\cr 0}, \qquad\hbox{and}\qquad
\lim_{\tau\to\infty}\exp(-\tau/4)\ffi_2(\tau)=\pmatrix{0\cr 1}. $$
Hence (2.6) has solutions of the form
$$y_1=\rme\left(\pmatrix{1\cr 0}+o(1)\right),\qquad
y_2= \ret\left(\pmatrix{0\cr 1}+o(1)\right), \eqno(2.8)$$
and therefore (2.1) has solutions $\psi_1$, $\psi_2$ such that
$$\psi_1(r)=\rme(\cos r+o(1)),\qquad \psi_2(r)=\ret(\sin r+o(1)),$$
$$\psi_1'(r)=-\rme(\sin r+o(1)),\qquad\hbox{and}\qquad \psi_2'(r)
=\ret(\cos r+o(1)).$$
Changing the signs that need
to be changed, we see that the same result is true when $\gamma < 0$, and
thus ii) of Theorem 2.1 follows. $\epf$
\medskip
{\it Remark.} Clearly an analogous result holds in
a neighborhood of $-\infty$.
\vskip 1cm
\noindent{\bf 3. Proof of the main result}
\vskip.3cm
First we prove Theorem 1.2 and then Theorem 1.1.
\medskip
{\it Proof of Theorem 1.2.} Clearly we may assume $\gamma>0$.
Let $\mag$ be as in the statement of Theorem 1.2. First we show that
$\mag$ is nonempty. By Theorem 2.1, for any given $V\in\lo$ we may
choose nonzero solutions $\psi_-$ and $\psi_+$ of
(1.4) with $k=\alpha/2$, such that
$\psi_{-}(r)$ goes to zero as $r$ goes to $-\infty$, and $\psi_{+}(r)$
goes to zero as $r$ goes to $+\infty$. By the same theorem we can also
choose $a>0$ such that
$\psi_{-} (-a) \psi_{+}(a)>0$. Now, if we define $\psi(r)=\psi_{-}(r)$ for
$r\leq-a$, $\psi(r)=\psi_{+}(r)$ for $r\geq a$, and
$\psi(r)=\varphi(r)$ for $|r|\leq a$, where $\varphi$ is any
$C^2$ function of constant sign that smoothly joins $\psi_-$ and
$\psi_+$ on $[-a, a]$, and set
$\tilde V(r)=V(r)$ for $|r|\geq a$ and
$\tilde V(r)=(\alpha^2/4)- (\gamma/r)\sin \alpha r+\varphi''/\varphi$
for $|r|\leq a$, then $\tilde V\in\lo$ and $\psi$ is a
nonzero continuously differentiable function which goes to zero
as $|r|$ goes to infinity and satisfies
$$-\psi'' + \gamma {\sin \alpha r\over r} \psi + \tilde V \psi =(\alpha^2/4)
\psi,\qquad\quad\hbox{a.e in $\real$.}$$
Hence $\tilde V\in\mag$.
In addition, it follows from the construction of $\tilde V$ that $\mag$
is unbounded in $\lor$. Furthermore,
if $V$ belongs to $\mag$, then in view of Theorem 2.1 we must have
$k=\alpha/2$.
\medskip
To complete the proof of Theorem 1.2 we only need to show that
$\mag$ is a smooth
sub-manifold of $\lo$ of
codimension one. This is proved in the following lemma.
\medskip
{\bf Lemma 3.1.} {\sl Let $\mag$ be as in Theorem 1.2. Then there exists a
$C^\infty$
function $F : \lor \longrightarrow \real$ such that zero is a regular value
of $F$ and $\mag = F^{-1}( \{ 0\}) $. }
\medskip
{\it Proof.}
For every $V\in\lo$ let $\psi_+$ be the solution of
$$-\psi'' + \gamma {\sin \alpha r \over r}\psi + V
\psi =(\alpha^2/4) \psi, \qquad r\in \real,
\eqno(3.1)$$
which coincides for large positive $r$ with the function $\phi$
given in (ii) of Theorem 2.1. First we will show that
$\psi_+$ and $\psi_+'$ depend smoothly on $V$.
For $r_0\in\real$, let $X_{r_0}$ be the Banach space of
continuous functions $\varphi$ from $[r_0,\infty)$ into $\real^2$
with the norm
$$ \| \varphi \|_{r_0} \equiv \sup_{\tau\geq r_0} \|\varphi(\tau)
\exp\left(\tau/ 4\right) \| < \infty. $$
It is easy to verify that the smoothness of $\psi_+$ and $\psi_+'$
with respect to $V$ follows from the
fact that for all $r_0\in\real$, the solution
$\varphi_1$ of (2.7) that satisfies
$$ \lim_{\tau\to +\infty} \exp(\tau/ 4)\ffi_1
=\pmatrix{1\cr 0}, $$
is a smooth a function of $L
\in \lor$ with values in $X_{r_0}$.
To prove this last define $\Phi:
L^1(\real)\times X_{\tau_0} \to X_{\tau_0}$ as
$$\Phi(L, \varphi)(\tau) =
\varphi(\tau) - \psi_1(\tau)
+ \int_\tau^\infty \Psi(\tau) \Psi^{-1}(s) L(s) \varphi(s) ds, $$
where, $\tau_0$ is as in (2.7),
$\displaystyle{\psi_1(\tau)=\exp(-\tau/ 4)\pmatrix{1\cr 0}}$,
and
$$ \Psi(\tau)=\pmatrix{ \exp (-\tau/4) & 0 \cr 0 & \exp (\tau/4)}.$$
Note that $\Phi(L,\varphi)=0$
if and only if $\varphi=\ffi_1$.
Next we fix $L_0 \in L^1(\real)$ and let $\varphi_0 \in
X_{\tau_0}$ be
so that $\Phi(L_0, \varphi_0)=0$. We prove that if
$\tau_0$ is sufficiently large, then $\Phi(L,\varphi)=0$
implicitly defines $\varphi_1$ as a smooth function of $L$, with
values in $X_{\tau_0}$, on a
neighborhood of $L_0$. Since $\Phi (\cdot,\cdot)$ is jointly smooth,
by the implicit function theorem it suffices to show
that for $\tau_0$
sufficiently large the operator $d_2 \Phi(L_0, \varphi_0)$ is
invertible from $X_{\tau_0}$ onto $X_{\tau_0}$.
Clearly
$$ (d_2 \Phi(L_0, \varphi_0))( H )= H + P(H), \qquad \hbox{ for all $H
\in X_{\tau_0}$,} $$
where
$$ P(H)(\tau) \equiv \int_\tau^\infty \Psi(\tau) \Psi^{-1}(s)
L_0(s) H(s) ds.$$
Using the definition of $\Psi$ it is easily
verified that
$$ \| P(H) \|_{\tau_0} \leq \| H \|_{\tau_0} \int_{\tau_0}^\infty
\| L_0(s) \| ds,
\qquad \hbox{ for all $H\in X_{\tau_0}$,} $$
from which the invertibility of $d_2 \Phi(L_0, \varphi_0)$
for large $\tau_0$ follows. Thus $\ffi_1$ is smooth in $L$
in a neighborhood ${\cal O}$ of $L_0$ in the
Banach space $X_{\tau_0}$. Since $\ffi_1(\tau_0)$ is smooth in $L$,
the smoothness of $\ffi_1$ as a function from
${\cal O}$ to $X_{r_0}$ follows from the fact that the solutions to
the initial value problem
$${d\ffi\over d\tau}=(G_1+L)\ffi,\qquad\qquad\ffi(\tau_0)=\ffi_1(\tau_0), $$
are smooth in $L$ and the initial value $\ffi_1(\tau_0)$. This last can
be proved in the usual way using the integral equation.
\medskip
Analogous arguments show that the solution
$\psi_-$ of (3.1) that satisfies
$$\psi_-(r)=|r|^{-\gamma/2\alpha} ( \cos kr + o(1) )
\qquad \hbox{ and } \qquad
\psi_-'(r)=- k|r|^{-\gamma/2\alpha} ( \sin k r + o(1) ), $$
as $r \to -\infty$, is a smooth function of $V$ and so is $\psi'_-$.
\medskip
Now we define $F:L^1(\real) \to \real $ as
$$F(V)=\left| \matrix{ \ppv& \pmv \cr \ppp&\pmp } \right|.$$
This function is well defined since the Wronskian of any pair of
solutions of (3.1) is
constant as a function of $r$.
Moreover $V \in \mag$ if and only if $F(V)=0$; or equivalently,
if and only if
$\psi_-= \lambda \psi_+$, where $\lambda \not= 0$ is a function of
$V$, constant as a function of $r$.
Thus $\mag=F^{-1}(\{0\})$.
Since $F$ is a smooth function of $V$,
to finish the proof it remains to show that zero
is a regular value of $F$, that is to say that for every
$V\in\mag$ we have $dF(V)\not= 0$.
Differentiating $F$ with respect to $V$ we find that for every $V$
and $H$ in $L^1(\real)$ we have
$$dF(V) (H) = \left| \matrix{ d\ppv (H) & \pmv \cr d\ppp (H) & \pmp
\cr} \right| + \left| \matrix{ \ppv & d\pmv (H) \cr \ppp &
d\pmp (H) \cr} \right|, \eqno(3.2) $$
where $d$ indicates differentiation with
respect to $V$ and the prime differentiation with respect to $r$,
with $V$ fixed.
In order to prove that $dF(V)$ is not zero we note first that,
for any fixed $a\in\real$,
$$\psi_+'(r,V)=\psi_+'(a,V)+\int_a^r\Lambda(t,V)\psi_+(t,V)dt, $$
where $\Lambda(r,V) \equiv (\gamma / r) \sin \alpha r + V-(\alpha^2/4)$.
Using the fact that for any interval $[c,d]$
the map $ V\mapsto \psi_+(\cdot, V) $, from $\lo$ to the
space $C[c,d]$ is smooth we have
$$ d\psi_+'(r,V)(H)=d\psi_+'(a,V)(H)+\int_a^r H(t)\psi_+(t,V)dt
+\int_a^r \Lambda(t,V)d\psi_+(t,V)(H)dt. $$
Thus $d\psi_+'(r,V)(H)$ is absolutely continuous with derivative
$$(d\psi_+'(r,V)(H))'=H(r)\psi_+(r,V)+\Lambda(r,V)d\psi_+(r,V)(H)\qquad\quad
\hbox{a.e.} $$
Now for any fixed $b\in\real$ and $\beta\geq b$ we consider
$$\eqalign{\int_b^\beta\psi_+(t,V)(d\psi_+'(t,V)&(H))'dt\cr
=&\int_b^\beta(\psi_+(t,V))^2H(t)+
\Lambda(t,V)\psi_+(t,V)d\psi_+(t,V)(H)dt. }$$
Integrating by parts we also have
$$ \eqalign{\int_b^\beta\psi_+(t,V)(d\psi_+'(t,V)&(H))'dt\cr
=&d\psi_+'(t,V)(H)\psi_+(t,V)\bigg|_b^\beta
-\int_b^\beta\psi_+'(t,V)d\psi_+'(t,V)(H)dt. } $$
Since $\psi_+(t,V)$ is not
a jointly $C^2$-function of $t$ and $V$, in order to perform
another integration by parts we show first
that for any $t\in\real$ and $V$ and $H$ in $\lo$ we have
$$ d\psi_+'(t,V)(H)=(d\psi_+(t,V)(H))'. \eqno(3.3)$$ To prove (3.3)
just note that
$\displaystyle{\psi_+(t,V)=\psi_+(c,V)+\int_c^t\psi'_+(\tau,V)d\tau}$.
Since $\psi'_+(\cdot,V)$ is smooth in $V$ as a function in $C[c,d]$
for any $d>c$,
we can differentiate with respect to $V$ under the integral sign and obtain
$$ d\psi_+(t,V)(H)=d\psi_+(c,V)(H)+\int_c^t d\psi_+'(\tau,V)(H)d\tau, $$
where for fixed $V$ and $H$, $d\psi_+'(\tau,V)(H) $ is continuous in $\tau$.
Thus (3.3) follows immediately.
So another integration by parts yields
$$ \eqalign{ \int_b^\beta(\psi_+(t,V))^2H(t)+&
\Lambda(t,V)\psi_+(t,V)d\psi_+(t,V)(H))dt\cr
= & \psi_+(t,V)d\psi_+'(t,V)(H) \bigg|_b^\beta
-\psi_+'(t,V)d\psi_+(t,V)(H)\bigg|_b^\beta\cr
&\qquad +\int_b^\beta\Lambda(t,V)
\psi_+(t,V)d\psi_+(t,V)(H)dt,} $$
which gives
$$\int_b^\beta\psi_+(t,V)^2H(t)dt=\left(\psi_+(t,V)d\psi_+'(t,V)(H)-
\psi_+'(t,V)d\psi_+(t,V)(H) \right)\bigg|_b^\beta. $$
Taking $\beta$ to infinity and utilizing the fact that for
fixed $V$ and $H$, the functions $\psi_+$, $\psi_+'$,
$d\psi_+$, and $d\psi_+'$ all
approach zero at infinity we obtain
$$\int_b^\infty\psi_+(t,V)^2H(t)dt=\psi_+'(b,V)d\psi_+(b,V)(H)-
\psi_+(b,V)d\psi_+'(b,V)(H). $$
\medskip
Analogously,
$$
\int_{-\infty}^b \psi_-(t,V)^2 H(t) dt
= \psi_-(b,V) d\psi_-'(b,V)(H) - \psi_-'(b,V) d\psi_-(b,V)(H).$$
Finally, combining (3.2) with these last two identities and using the
fact that for $V \in \mag$
we have
$\psi_-=\lambda \psi_+ $, $\lambda\not =0$, we find that
$$ \eqalign{
dF(V)(H) & = \lambda \int_b^\infty \psi_+(t,V)^2 H(t) dt
+ {1 \over \lambda} \int_{-\infty}^b \psi_-(t,V)^2 H(t) dt \cr
&= \int_{-\infty}^\infty \psi_-(t,V)\psi_+(t,V) H(t) dt, } $$
for all $H\in\lo$. Therefore, if $V\in\mag$ then $dF(V)$
is the linear functional on $\lo$ defined by the function
$\psi_-\psi_+\in
L^\infty(\real)\setminus\{0\}$. $\epf$
\medskip
{\it Proof of Theorem 1.1.} For $p\geq 1$, $\alpha>0$,
and $\gamma\in\real\setminus \{0\}$,
let $\mpg$ be the set of functions $V\in\lo$ for which the operator
$\hvp$ has a positive eigenvalue. It follows from Theorem 2.1 and
Proposition A.1 that
$$ \mpg=\cases{ \emptyset, & if $ \plt$, \cr \mag, &if $\pgt$.}
\qquad\qquad \epf $$
\vskip.5cm
\noindent{\bf 4. Appendix }
\vskip.3cm
Here we establish the connection between the eigenfunctions
of the operator $\hvp$ and the decaying solutions of (1.4).
Below we use {\it Duhamel's formula} [27] in the form
$$(\ffi, \uvt f)=(\ffi, \uzv f)
-\int_0^t(U_Q(t-u)\ffi, Q U_0(u) f) du, \eqno({\rm A.1}) $$
for $\ffi\in\test$ and $f\in L^\infty(\real)\cap \lp$, where
$Q$ is as in (1.1) and
$\uvt$ is as introduced in (1.2). Formula (A.1) is readily
established by an approximation argument
starting with bounded $Q$.
Here
$$ (\phi,\psi)=\imi\overline{\phi(x)}\psi(x) dx.$$
The main result of this section is
\proclaim Proposition A.1. Let $p$, $Q$, and $\hvp$ be as in (1.1). Then
$f\in\lp$ is an eigenfunction of $\hvp$ corresponding
to the eigenvalue $\lambda\in\real$ if and only if $f$ is a differentiable
function that vanishes at infinity, such that $f'$ is
absolutely continuous
on every finite interval of $\real$ and
$$ -f''+Qf=\lambda f\qquad\qquad\hbox{a.e.} \eqno({\rm A.2})$$
{\it Proof.}
Suppose $f\in\lp$ is a differentiable function that vanishes
at infinity, such that $f'$ is absolutely continuous on every
finite interval of $\real$ and that (A.2) is satisfied.
We will show that
$$ \upq f=\exp(-\lambda t) f,\qquad\quad t\geq 0.\eqno({\rm A.3})$$
For any given $\ffi\in C_0^\infty$ we define
$$ g(s)=(\ffi,U_Q(s) f),\qquad\qquad s\geq 0. $$
We show first that for any $s\geq 0$ we have
$$ \lim_{t\to 0^+}{g(s+t)-g(s)\over t}=-\lambda g(s). $$
Set $\psi=U_Q(s)\ffi$. Using Duhamel's formula and the fact [6] that
$(\phi,U_Q(s)\xi)=(U_Q(s)\phi, \xi)$, for $\phi\in\lp$ and
$\xi\in L^{p'}(\real)$, we find that
$$\eqalign{ {g(s+t)-g(s)\over t}=&
\left( \psi, {\uvt -1\over t}f \right)\cr
=& \left( \psi, {\uzv -1\over t}f
\right)-{1\over t}\int_0^t(U_Q(t-u)\psi,
Q U_0(u)f)du. }\eqno ({\rm A.4})$$
We show next that as $t\to 0^+$ the right side of (A.4) goes to
$(\psi,f'') -(\psi, Qf) =-\lambda g(s)$.
In fact, using the kernel
$p_t(x,y)$ of $\uzv$ introduced in (1.3) we have
$$\eqalign{ \left( \psi, {\uzv-1\over t}f \right)=&
\left( \psi, {1\over t}\imi p_t(x,y)(f(y)-f(x))dy \right) \cr
=&\left( \psi, {1 \over t}\imi p_t(0,y)(f(x+y)-f(x))dy\right)\cr
=&\left(\psi,{1\over t} \imi p_t(0,y)\int_0^y(y-u)f''(x+u)du dy \right)\cr
= & \left( \psi, f''+\imi p_t(0,y)
\int_0^y (y-u){ (f''(x+u)-f''(x))\over t}du dy\right), }$$
where in the third equality we have used Taylor's formula
$$f(x+u)-f(x)=yf'(x)+\int_0^y(y-u)f''(x+u)du. $$
Setting $z=y/\sqrt t$ and then $u=\sqrt t w$ we find that
$$\eqalign{ \imi p_t(0,y)
\int_0^y (y-u) &{ (f''(x+u)-f''(x))\over t}du dy\cr
=& \imi p_1(0,z)\int_0^z(z-w)(f''(x+\tau w) -f''(x))dw\,dz,} $$
where $\tau\equiv \sqrt t$. Hence
$$\eqalign{ \bigg|\bigg( \psi, \imi & p_t(0,y)
\int_0^y (y-u){ (f''(x+u)-f''(x))\over t}du dy\bigg) \bigg|\cr
\leq & {1\over\sqrt{4\pi}}\imi |\psi(x)|\imi \exp(-z^2/4) \int_{-|z|}^{|z|}
2|z| |f''(x+\tau w)-f''(x)|dw\,dz\,dx\cr
\leq & {1\over\sqrt{4\pi}}\imi|\psi(x)|\imi\int_{|w|}^\infty
2z \exp(-z^2/4)|f''(x+\tau w)-f''(x) |dz\,dw\,dx \cr
=& {2\over\sqrt \pi}\imi|\psi(x)|\imi \exp(-w^2/4)
|f''(x+\tau w)-f''(x)|dw\,dx } $$
Thus, using (A.2), the fact that $f\in \lp\cap L^\infty(\real)$,
and the dominated
convergence theorem we see that the right side of the last inequality
goes to zero as $t\to 0^+$ and therefore
$$ \lim_{t\to 0^+}\left( \psi, {\uzv-1\over t}f \right)=(\psi,f''). $$
By the continuity of the function $(U_Q(u)\psi, Qf)$ with respect to $u$,
the second term on the right side of (A.4) approaches $-(\psi,Qf)$ since
$${1\over t}\int_0^t\left(U_Q(t-u)\psi, Q(U_0(u)-1)f\right)du $$
goes to zero as $t\to 0^+$.
To see this last we use the fact that
$U_Q(t)$ maps $L^\infty(\real)$ into $L^\infty(\real)$ and that
$\displaystyle{\|U_Q(t)\|_{L^\infty\rightarrow L^\infty }\leq C}$,
for small $t$,
that $Q=W+V$, with $V\in\lo$
and
$W\in \lp$, for $p>1$, and that $(\uzv -1)f$
converges uniformly to zero as $t\to 0^+$ since $f$ vanishes
at infinity and hence
is uniformly continuous on $\real$.
Thus we have proved that the right derivative $D_+g $ of
the function $g(s)$ satisfies
$ D_+g(s)=-\lambda g(s)$,
for all $s\geq 0$. It follows that $ D_+(\exp(\lambda s)g(s))=0$ for
$s\geq 0$ and therefore [24] that $g(s)=\exp(-\lambda s)g(0)$,
which proves (A.3).
Suppose now that $f\in\lp$ satisfies (A.3).
Then [6, 26]
$f\in L^\infty(\real)$, $f$ is continuous, vanishes at infinity,
and $f'$ exists and belongs to
$ L^2_{\rm loc}(\real)$. By Duhamel's formula, for every
$\ffi\in \test$ we have
$$(\ffi,\uvt f)=(\ffi, \uzv f)-\int_0^t\left(U_Q(t-u)\ffi,
Q U_0(u) f \right)du. $$
Differentiating this last at $t=0$, using (A.3),
we obtain
$-\lambda(\ffi,f)=(\ffi'', f)-(\ffi, Qf) $
and thus
$ (\ffi',f')=(\ffi,(\lambda -Q)f)$ for all
$\ffi\in\test$. Standard approximation arguments show that $f'$
is almost everywhere equal to an absolutely continuous function and
that (A.2) is satisfied. $\epf$
\vfill\eject
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\end