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\begin{document}
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\begin{center}
{\large \bf Eigenvalues imbedded in the band spectrum
for the periodic array of quantum dots}\\
\bigskip
{\large V.~A.~Geyler${}^{a)}$,\, I.~Yu.~Popov${}^{b)}$}\\
\bigskip
{\it ${}^{a)}$Department of Mathematics, Mordovian State University,
Saransk 430000, Russia\\
\medskip
${}^{b)}$Department of Higher Mathematics, Institute of Fine
Mechanics and
Optics, Sablinskaya 14, St.-Petersburg, 197101, Russia}
\end{center}
\bigskip
\noindent{\bf Abstract}
\bigskip
Solvable model of a periodic array of quantum dots in a uniform
magnetic field is proposed having eigenvalues imbedded in the
continuous
spectrum. The continuous spectrum of the array may be absolutely
continuous as well as singular continuous one. The model is
based on the operator extension theory.
\newpage
\noindent{\bf I. Introduction}
\medskip
The problem of eigenvalues imbedded in the continuous spectrum is
of
great interest both from mathematical and physical points of view.
Mathematicians are studying it actively because there is no general
theory allowing one to describe situation effectively, and to develop
it
they are trying to reveal general features by studying concrete
problems [1].
An important reason for physicists is that the problem in question is
closely related with the description of transport properties of many
physical systems in fluid mechanics, acoustics, and quantum
mechanics [2--4].
Among quantum transport problems great attention is attracted to
unusual phenomena in such quantum nanostructures as arrays of quantum
dots and antidots [5--9], espesially subjected to magnetic fields.
The magnetic field may result in such an exotic effect as an
appearence of a fractal structure of the energy spectrum predicted in
[10--12]. On the other hand,
it is known that magnetic field leads to the localization effects in
periodic
systems. This phenomena was discovered by Ando [13] in numerical
experiments. Later the existence of the localizations was proved
rigorously [14-17].
One can expect that there may be eigenvalues imbedded in
the continuous spectrum in such systems. In our paper this
effect is considered in the framework of the model based on the
operator
extension theory. The model was proposed and studied on the physical
level of rigour in [18, 19].
The main goal of the present paper is to prove
rigorously the existence of eigenvalues imbedded in the continuous
spectrum of the model Hamiltonian for an appropriate choice of the
model
parameters. We stress that the continuous spectrum in question may be
singular as well as absolutely continuous. As for the physical
motivation,
our model is based on physical assumptions like those in the Hubbard
model
(see [19] for details).
The paper is organized as follows. In the beginning we give a
construction
of the model. The essential point is that we use a variant of the
extension theory model which is known as "zero-range potentials with
internal structure" [20, 21]. The main result is contained in the
third
section where we study the band structure of the spectrum. In the
particular case of the mono-atomic square lattice we
reduce the spectral problem for the model Hamiltonian to the spectral
problem for the well-known Harper operator [11, 12].
Note that proofs presented here are valid also
for arbitrary lattice including multi-atomic one.
\bigskip
\noindent{\bf II. Model}
\medskip
We consider an array of quantum dots displaced in the nodes of a plane
lattice $\Gamma$ which is invariant with respect to translations by
vectors
of a Bravais lattice $\Lambda$. We assume that ${\bf 0} \in \Gamma$
and
fix some primitive vectors ${\bf a}_1$, ${\bf a}_2$ of $\Lambda$. The
corresponding unit cell is marked as $F_{\Lambda}$:
%
\begin{equation}\label{0}
F_{\Lambda} = \{t_1 {\bf a}_1 +t_2 {\bf a}_2 : 0 \leq t_1, t_2 < 1 \}.
\end{equation}
%
Let ${\rm K}$ be the set of all points of $\Gamma$ belonging to
$F_{\Lambda}$.
Thus each point $\gamma$, $\gamma\in\Gamma$, has the unique
representation
in the form $\gamma=\lambda+\kappa$ where $\lambda\in\Lambda$,
$\kappa\in
{\rm K}$.
The Hamiltonian $H_d$ of spinless charged particle in a single
quantum dot
will be chosen in the form $H_d=H_0+V(r)$ where $V(r)$ is the
confining
potential of the dot and $H_0$ is free particle
Hamiltonian with the presence of a magnetic field, i.e. the
Schr\"odinger
operator in $L^2({\bf R}^2)$ defined by the differential expression
%
\begin{equation}\label{00}
H_0=\left(-i\frac{\partial}{\partial x}-\pi\xi y\right)^2+
\left(-i\frac{\partial}{\partial y}+\pi\xi x\right)^2
\end{equation}
%
(which is known also as the Landau operator).
Here $\xi$ is the flux density mesured in the magnetic flux quanta
$\Phi_0$,
$\Phi_0=(2\pi\hbar c)^{-1}e$, $x$ and $y$ are Cartesian coordinates
on the
plane. We choose the confining potential in a parabolic form:
$V(r)=\omega_0^2r^2/4$.
Below we need the description of the spectrum and eigenfunctions
of the operator $H_d$. To describe them let us
introduce the notations: $\omega_c=4\pi\vert\xi\vert$ ($\omega_c$ is
known as the cyclotron frequency),
$\Omega=\sqrt{\omega_c^2+4\omega_0^2}$,
and $\omega_{1,2}=(\Omega\pm\omega_c)/2$ (the hybrid frequencies).
The spectrum of $H_d$ is pure point and consists of the eigenvalues
known as the ``Fock--Darwin'' levels [23]:
%
\begin{equation}\label{0.0}
E_{mn}=\frac{\omega_c}{2}m+ \frac{\Omega}{2}(2n+|m|+1),
\qquad m\in\ZZ,\quad n\in\NN
\end{equation}
or in a more symmetric form
%
\begin{equation}\label{0.1}
E_{n_1n_2}=\omega_1(n_1+1/2)+\omega_2(n_2+1/2)
\end{equation}
%
where
%
$$
n_1=n+\frac{|m|+m}{2},\quad n_2=n+\frac{|m|-m}{2}.
$$
%
Using the polar coordinates $(\rho,\,\phi)$, we have the following
representation of the corresponding eigenfunctions $\Psi_{mn}$
%
\begin{equation}\label{0.2}
\Psi_{mn}=
c_{mn}\,e^{im\phi}\,\rho^{\vert m\vert}\,
\exp\left(-\rho^2/4l_0^2\right)\,
L^{\vert m\vert}_n\left(\rho^2/2l^2_0\right).
\end{equation}
%
Here $l_0=(2/\Omega)^{1/2}$ is the so-called magnetic length,
$L_n^k(x)$ are the generalized Laguerre
polynomials, and $c_{mn}$ are norming constants.
The state space ${\cal H}$ of our model is the direct sum
%
\begin{equation}\label{1}
{\cal H} = \sum^{}_{\gamma\in\Gamma}{}^{\oplus}{\cal H}_{\gamma}
\end{equation}
%
where each space ${\cal H}_{\gamma}$ is
the state space $L^2({\bf R}^2)$ of $H_d$.
Assuming that the tunneling between dots is absent, we come to the
following
"background" Hamiltonian $H^0_{arr}$:
%
\begin{equation}\label{0.3}
H^0_{arr}=\sum^{}_{\gamma\in\Gamma}{}^{\oplus}H_{\gamma}
\end{equation}
%
where each operator $H_{\gamma}$ is equal to $H_d$.
To take into account the charge carrier tunneling
between the dots we use the ``restriction -- extension procedure'' of
the
operator extension theory [20--22].
Let ${\cal D}_d$ be the set of all the
functions $f$ from the domain $D(H_d)$ each of which vanishes
in a neighbourhood of zero.
We denote by $S_d$ the symmetric operator in $L^2(\RR^2)$ which is the
restriction of $H_d$ to the domain ${\cal D}_d$. Let $S_{\gamma}=S_d$
for
each $\gamma\in\Gamma$ and
$S=\sum{}^{\oplus}_{\gamma\in\Gamma}S_{\gamma}$. We shall
seek the ``true'' Hamiltonian $H$ of the quantum dot array among
self-adjoint extensions of $S$. Let $R(\zeta)$ be the
resolvent of the desired operator: $R(\zeta)=(H-\zeta)^{-1}$, and let
$R^0(\zeta)=(H^0_{arr}-\zeta)^{-1}$. Then $R$ and $R^0$ are related
by means of
the Krein resolvent formula [24]:
%
\begin{equation}\label{2}
R(\zeta)=R^0(\zeta)-{\rm B}(\zeta)[Q(\zeta)+A]^{-1}{\rm
B}^*(\bar\zeta).
\end{equation}
%
Here the operator valued holomorphic functions ${\rm B}(\zeta)$ and
$Q(\zeta)$ are so-called Krein $\Gamma$- and ${\cal Q}$-functions
respectively, and $A$ is a self-adjoint operator in the deficiency
space
of $S$ which parametrizes the self-adjoint extensions of $S$. In the
considered case we can rewrite Eq.(\ref{2}) in the following more
detailed form. In virtue of (\ref{1})
%
\begin{equation}\label{0.4}
R^0 (\zeta) = \sum_{\gamma\in\Gamma}{}^{\oplus}R_{\gamma}(\zeta)
\end{equation}
%
where each $R_{\gamma}(\zeta)$ is the resolvent of the operator $H_d$.
Due to the same formula (\ref{1}) we can get the representations:
%
\begin{equation}\label{0.5}
{\rm B}(\zeta)= \sum^{}_{\gamma\in\Gamma}{}^{\oplus}{\rm
B}_{\gamma}(\zeta),
\quad Q(\zeta) = \sum_{\gamma\in\Gamma}{}^{\oplus} Q_{\gamma}(\zeta)
\end{equation}
%
where for each $\gamma\in\Gamma$ ${\rm B}_{\gamma}(\zeta)$
(respectively $Q_{\gamma}(\zeta)$) coincides with the Krein
$\Gamma$-function ${\rm B}_d(\zeta)$ (respectively, with the
${\cal Q}$-function $Q_d(\zeta)$)) of the pair $(H_d,\,S_d)$.
Identifying the deficiency subspace of $S_{\gamma}$ with the standard
one-dimensional space ${\bf C}$, we obtain the operator
${\rm B}_{\gamma}(\zeta)$ as an operator from ${\bf C}$ into
$L^2({\bf R}^2)$ which acts as multiplication by the function
$G_d(x,0;\zeta)$ [25]
where $G_d(x,y;\zeta)$ is the Green function of $H_d$,
i.e. the integral kernel of its resolvent $R_d(\zeta)$. Thus the
operator
${\rm B}_d(\zeta){\rm B}_d^*(\bar\zeta)$ is an integral operator in
$L^2({\bf R}^2)$ with the kernel
$B_d(x,y;\zeta)=G_d(x,0;\zeta)G_d(0,y;\zeta)$.
The complex-valued function $Q_d(\zeta)$ can be
obtained up to an additive constant by the formula [25]:
%
\begin{equation}\label{2a}
Q_d(\zeta) =
\lim_{x\to 0} (G_d(x,0;\zeta) - G_d(x,0;\zeta_0))
\end{equation}
%
where $\zeta_0$ is some fixed regular point of the operator $H_d$.
To get $G_d$ one can use the Laplace transform of time-dependent
Green function $K_d(x,y;t)$ (i.e. the integral kernel of the
propagator of
$H_d$). This kernel has been obtained in [26]:
%
$$
K_d(x,y;t)=\frac{\Omega}{8\pi i\sin\frac{\Omega t}{2}}\times
$$
%
\begin{equation}\label{3}
\exp\left\{\frac{i\Omega}{4\sin\frac{\Omega t}{2}}\left[
(x_2y_1-x_1y_2)\sin\frac{\omega_c t}{2}-
(x_1y_1+x_2y_2)\cos\frac{\omega_c t}{2}+
\frac{1}{2}(x^2+y^2)\cos\frac{\Omega t}{2}\right]\right\}.
\end{equation}
%
In particular, the time-dependent Green function $K_0$ for the Landau
operator has the form:
%
$$
K_0(x,y;t)=\frac{\omega_c}{8\pi i\sin\frac{\omega_c t}{2}}\times
$$
%
\begin{equation}\label{4}
\exp\left\{\frac{i\omega_c}{4\sin\frac{\omega_c t}{2}}\left[
(x_1y_2-x_2y_1)\sin\frac{\omega_c t}{2}-
(x_1y_1+x_2y_2)\cos\frac{\omega_c t}{2}+
\frac{1}{2}(x^2+y^2)\cos\frac{\omega_c t}{2}\right]\right\}.
\end{equation}
%
If ${\rm Im}\,\,\zeta>0$, then $G_d(x,y;\zeta)$ is the Laplace
transform of
$K_d(x,y,;t)$, namely
%
\begin{equation}\label{0.6}
G_d(x,y,;\zeta)=-i\int_0^{\infty}K_d(x,y;t)\,e^{it\zeta}\,dt\,;
\end{equation}
%
the relation between $G_0$ and $K_0$ is analogous to Eq.(\ref{0.6}).
In the case of $\omega_0=0$ (i.e. in the case of the Landau operator
$H_0$) the Krein ${\cal Q}$-function $Q_0(\zeta)$ is known and has the
form [15]:
%
\begin{equation}\label{5}
Q_0(\zeta)= -\frac{1}{4\pi}[\psi(1/2-\zeta/\omega_c)+
\log(\omega_c/4\pi)+2C_{Euler}]
\end{equation}
%
where $\psi$ is the logarithmic derivative of the
Euler $\Gamma$-function and $C_{Euler}$ is the Euler constant.
Comparing (\ref{3}), (\ref{4}), and (\ref{0.6}) one can see that
(\ref{5}) and (\ref{2a}) result in
%
\begin{equation}\label{6}
Q_d(\zeta)= -\frac{1}{4\pi}[\psi(1/2-\zeta/\Omega)+\log(\Omega/4\pi)
+2C_{Euler}].
\end{equation}
%
It is convenient to rewrite Eq.(\ref{2}) representing the state
space ${\cal H}$ in the form of tensor product
${\cal H}=L^2({\bf R}^2)\otimes l^2(\Gamma)$. Then, obviously,
$R^0(\zeta)=
R_d(\zeta)\otimes I$ and
${\rm B}(\zeta)[Q(\zeta)+A]^{-1}{\rm B}^*(\bar\zeta)$ has the form
$B_d(\zeta)\otimes T(\zeta)$ where
$T(\zeta)$ is an operator in $l^2(\Gamma)$ having a diagonal matrix
$T_{\gamma\gamma'}(\zeta)=Q(\zeta)\delta_{\gamma\gamma'}$.
The operator
$A$ must be invariant with respect to a unitary representation of the
magnetic translation group [27] in the space $l^2(\Lambda)$
(see [15]) and hence the
matrix $A_{\gamma\gamma'}$ of the operator $A$ is fully determined
according
to the formula
$$
A_{\gamma+\lambda,\kappa'+\lambda}=
\exp[\pi i\xi(\lambda\wedge(\gamma-\kappa'))]A_{\gamma\kappa'}
$$
by the elements $A_{\gamma\kappa}$ ($\kappa\in{\rm K}$) only [18].
\bigskip
\noindent{\bf III. Structure of the spectrum}
\medskip
We start from a simple proposition.
{\bf Proposition 1.} {\it The Fock-Darwin levels $E_{mn}$ with $m\ne0$
are eigenvalues of $H$.}
This proposition is a consequence of the following statement:
{\bf Proposition 2.} {\it Let $\tilde H$ be any self-adjoint
extension of
$S_d$. The Fock-Darwin levels $E_{mn}$ with $m\ne0$
are eigenvalues of $\tilde H$ and the corresponding eigenvectors
are equal to $\Psi_{mn}$.}
{\large Proof}. The statement expresses a well-known fact that a point
perturbation leaves fixed the states with non-zero angular momentum.
However, as far as we known, there are no proofs of this assertion in
the
sufficiently general case. To prove the proposition we use the Krein
resolvent formula (\ref{2}) once again. According to it, the Green
function
$\tilde G(x,y;\zeta)$ of $\tilde H$ has the following form
%
\begin{equation}\label{3.1}
\tilde G(x,y;\zeta)=G_d(x,y;\zeta)-\left[Q_d(\zeta)+\alpha\right]^{-1}
G_d(x,0;\zeta)G_d(0,y;\zeta)
\end{equation}
%
where $\alpha$ is a real number. With the help of (\ref{3.1}) we can
obtain
the following explicit description of $\tilde H$. Let $\zeta_0$,
${\rm Im\,\,\zeta_0}\ne0$, be fixed. Then the domain $\tilde{\cal D}$
of
$\tilde H$ consists of all the functions $f\in L^2(\RR^2)$ having the
form
%
\begin{equation}\label{3.2}
f(x)=g(x)-\left[Q_d(\zeta)+\alpha\right]^{-1}g(0)G_d(x,0;\zeta)
\end{equation}
%
where $g$ is an arbitrary function from $D(H_d)$ (it is easy to show
that
$D(H_d)\subset C(\RR^2))$. For given $f$ such a function $g$ is
uniquely
defined, and
%
\begin{equation}\label{3.3}
\tilde H(f)=\zeta_0f+(H_d-\zeta_0)g.
\end{equation}
%
The formula (\ref{3.2}) and (\ref{3.3}) imply in view
of (\ref{0.2}) the statement of the proposition.
Before to study the continuous part of the spectrum of $H$ we note
that the spectral properties of a periodic operator with a
uniform magnetic field depend on whether geometric parameters
of the system are commensurable with the so-called magnetic length.
More
precisely, let $\eta$ be the number of magnetic flux quanta through
the unit cell $F_{\Lambda}$, i.e. $\eta=\xi\vert F_{\Lambda}\vert$
where $\vert F_{\Lambda}\vert$ is the area of $F_{\Lambda}$.
Then the spectral properties of $H$ drastically depend on whether
$\eta$
is a rational number or not [10--12, 29].
To avoid some technical difficulties connected with this circumstance
we restrict ourselves to the most interesting case being
simultaneously a relatively simple one. Namely, we shall consider
only the case when $\Lambda=\Gamma$ and the operator $A$ in (\ref{2})
has
the form $A=A'+A''$ where $A'$ is a scalar operator, i.e. an operator
with a diagonal matrix $A'_{\lambda\mu}=a'\delta_{\lambda\mu}$, and
$A''$
is a nearest-neighbour hopping operator. Thus matrix of $A''$ has the
form
%
\begin{equation}\label{3.4}
A''_{\lambda0}=a''[\delta(\lambda_1,0)(\delta(\lambda_2,1)+
\delta(\lambda_2,-1))
+\delta(\lambda_2,0)(\delta(\lambda_1,1)+\delta(\lambda_1,-1))].
\ \end{equation}
%
Here $a'$ and $a''$, $a''\ne0$,
are some real non-zero constants (coupling constants), and
we write $\delta(k,l)$ instead of
$\delta_{kl}$ for convenience. In this case we are able to describe
the continuous spectrum of $H$ completely [18, 19], some mathematical
details will appear in [28].
To do it we denote by $\chi(x)$ the
inverse function to $\psi(x)$, $x\in{\bf R}$; $\chi$ is a multivalued
real-analytic function defined for all $x$ in ${\bf R}$ having
continuous
single-valued
branches $\chi_n(x)$ ($n=0,1,\ldots$) with values in the intervals
$(0,+\infty)$, $(-1,0)$, $(-2,-1)$, ... . We denote by $\tau_n$ the
functions
%
\begin{equation}\label{3.5}
\tau_n(x)=\Omega[1/2-\chi_n(4\pi a''x+4\pi a'-\log(\Omega/4\pi)-
2C_{Euler})].
\end{equation}
%
Finally, let us consider the Harper operator,
i.e. the direct integral with respect to $\nu$, $\nu\in[0,\,2\pi)$,
of difference operators $h(\eta;\nu)$ which are defined
in the space $l^2({\bf Z})$ by the formula:
$$
(h(\eta)\phi)_m=\phi_{m+1}+\phi_{m-1}+2\cos(2\pi m \eta+\nu)\phi_m.
$$
Here $\phi=(\phi_m)_{m\in{\bf Z}}\in\l^2({\bf Z})$.
The following proposition take place [19, 28]:
{\bf Proposition 3}. {\it
The continuous spectrum of $H$ consists of the
magneto-Bloch bands $B_n$. The band $B_n$ is the set
of all values of the function $E=\tau_n(x)$ on the spectrum of the
Harper
operator $h(\eta)$; $B_n$ lies below the level $E_{0n}$ and (if
$n\geq 1$)
above the level $E_{0,n-1}$.}
{\bf Corollary}. {\it Let an eigenvalue $E_{mn}$ be distinct from
every
$E_{0k}$, $k\in\NN$. Then for every $a''\ne0$ there exists a coupling
constan
$a'$ such that $E_{mn}$ is imbedded in the continuous spectrum of
$H$.}
{\large Proof}. Since $\omega_c<\Omega$, the eigenvalue $E_{mn}$ lies
between some eigenvalues of the form $E_{0,n-1}$ and $E_{0n}$ where
$n\geq1$, i.e. between the levels $\Omega(n-1)/2$ and $\Omega n/2$.
On the other hand, the image of the function $\chi_n$ is the whole
interval
$(-n,-n+1)$. Fix $x$ in the spectrum of $h(\eta)$, then
according to (\ref{3.5}) $\tau_n(x)$ run from $\Omega(n-1)/2$
to $\Omega n/2$ as $a'$ run from $-\infty$ to $+\infty$; this proves
the collorary.
{\bf Remark}. We stress that the spectrum of $h(\eta)$ is singular if
$\eta$ is an
irrational number satisfying some arithmetic conditions [29, 30].
For such $\eta$ the continuous spectrum of $H$ is singular as well.
Hence we have an example of operators having eigenvalues imbedded into
the singular continuous spectrum.
\bigskip
\noindent{\bf Acknowledgements}
\medskip
The work is partly supported by the
RFBR grants, Soros Foundation and ANS RF.
\newpage
\begin{thebibliography}{99}
{\small
\bibitem{Nab} Naboko S.N. and Pushnitskii A.B.: {\it Funkt. Anal. i
Pril.}
{\bf 29} (1995), 31 (In Russian).
\bibitem{Eva} Evans D.V., Levitin M., and Vassiliev D.: {\it J.
Fluid Mech.}
{\bf 261} (1994), 21.
\bibitem{Par} Parker R. and Stoneman S.A.T.: {\it Proc. Inst. Mech.
Engrs.}
{\bf 203} (1989), 9.
\bibitem{Urs} Ursell F.: {\it Proc. R. Soc. London.} {\bf A435}
(1991), 574.
\bibitem{Wei} Weiss D., Roukes M.L., Menschig A., Grambow P.,
von~Klitzing K., and Weimann G.: {\it Phys. Rev. Lett.}
{\bf 66} (1991), 2790.
\bibitem{Lor} Lorke A., Kotthaus J.P., and Ploog K.: {\it Phys. Rev.}
{\bf B44} (1991), 3447.
\bibitem{Gus} Gusev G.M., Dolgopolov V.T., Kvon Z.D., Shashkin A.A.,
Kudryashov V.M., Litvin L.V., and Nastaushev Yu.V.:
{\it Pis'ma Zh. Eks. Teor. Fiz.} {\bf 54} (1991) 369
(In Russian).
\bibitem{Bas} Baskin E.M., Gusev G.M., Kvon Z.D., Pogosov A.G., and
Entin M.V.: {\it JETP Lett.} {\bf 55} (1992), 678.
\bibitem{Wei2} Weiss D., Richter K., Menschig A., Bergmann R.,
Schweizer H.,
von~Klitzing K., and Weimann G.: {\it Phys. Rev. Lett.}
{\bf 70} (1993), 4118.
\bibitem{Azb} Azbel M.Ya.: {\it Sov. Phys. ZETP.} {\bf 19} (1964),
634.
\bibitem{Hof} Hofstadter D.R.: {\it Phys. Rev.} {\bf B14} (1976),
2239.
\bibitem{Wan} Wannier G.H.: {\it Phys. Status Solidi.} {\bf B88}
(1978),
757.
\bibitem{And} Ando T.: {\it J. Phys. Soc. Japan.} {\bf 52} (1983),
1740.
\bibitem{GM} Geyler V.A. and Margulis V.A.: {\it Theor. Math. Phys.}
{\bf 58} (1984), 302.
\bibitem{Gey} Geyler V.A.: {\it St.~Petersburg Math. J.} {\bf 3}
(1992),
489.
\bibitem{Av1} Avishai Y., Redheffer R.M., and Band Y.B.: {\it J.
Phys.}
{\bf A25} (1992), 3663.
\bibitem{Av2} Avishai Y., Azbel M.Ya., and Gredescul S.A.: {\it
Phys. Rev.}
{\bf B48} (1993), 17280.
\bibitem{GP1} Geyler V.A. and Popov I.Yu.: {\it Zeitschr. Phys.}
{\bf B93} (1994), 437.
\bibitem{GP2} Geyler V.A. and Popov I.Yu.: {\it Zeitschr. Phys.}
{\bf B98} (1995), 473.
\bibitem{Pa1} Pavlov B.S.: {\it Theor. Math. Phys.} {\bf 59} (1984),
544.
\bibitem{Pa2} Pavlov B.S.: {\it Russian Math. Surv.} {\bf 42}
(1987), 127.
\bibitem{Alb} Albeverio S., Gesztesy F., H\o egh-Krohn R., and
Holden H.:
{\it Solvable Models in Quantum Mechanics}, Springer,
Berlin
(1988).
\bibitem{Foc} Fock V.: {\it Z. Phys.} {\bf 47} (1928), 446.
\bibitem{Kre} Krein M.G. and Langer H.: {\it Funkt. Anal. i Pril.}
{\bf 5} (1971), 54 (In Russian).
\bibitem{GMC} Geyler V.A., Margulis V.A., and Chuchaev I.I.: {\it
Sibir.
Matem. Zhurn.} {\bf 36} (1995), 823 (In Russian).
\bibitem{Pap} Papadopoulos G.J.: {\it J. Phys.} {\bf A4} (1971), 773.
\bibitem{Zak} Zak J.: {\it Phys. Rev.} {\bf 134} (1964), A1602.
\bibitem{GPP} Geyler V.A., Pavlov B.S., and Popov I.Yu.: {\it Atti
Sem.
Mat. Fis. Univ. Modena} (In press).
\bibitem{Hel} Helffer B. and Sj\"ostrand J.: {\it Bull. Soc. Math.
France.}
{\bf117}, Suppl. 396 (1989), 1.
\bibitem{Cho} Choi M.-D., Elliott G.A., and Yui N.: {\it Invent.
math.}
{\bf99} (1990), 225.
}
\end{thebibliography}
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