%Physics Letters A, 1996, V. 222, 286-290.
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\title{Eigenvalues and bands imbedded in the continuous spectrum
for the system of resonators and waveguide: solvable model}
\author{I.Yu.Popov, S.L.Popova\\
Department of Higher Mathematics,\\
Leningrad Institute of Fine Mechanics and Optics,\\
14 Sablinskaya, 197101 Leningrad, Russia, USSR.\\}
\begin{document}
\maketitle
\begin{abstract}
Solvable model based on the operator extension theory is suggested
for the description of trapped modes imbedded in the continuous
spectrum. A system of resonators connected through small apertures
with a waveguide is considered as for the case of Neumann and
Dirichlet boundary conditions. That is, we study both acoustical
and quantum waveguides. The existence of such modes is shown
(corresponding sufficient condition is derived). An effective and
simple algorithm for its determination is suggested.
A system of acoustic (quantum) waveguide and periodic set of coupled
cavities
is studied in the framework of the model. The dispersion equation
is obtained in an explicit form. The existence of bands imbedded in
the continuous spectrum is proved, and an algorithm for its
determination is described.
\end{abstract}
\newpage
The problem of eigenvalues imbedded in the continuous spectrum is
of
great interest now 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 of concrete
problems
(see, for example, ~\cite{NP}).
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 acoustics, quantum mechanics, fluid mechanics
~\cite{ELV}, ~\cite{PS}, ~\cite{U}. We shall deal with a system of
resonators connected with a waveguide. Note that the problem of
eigenvalues
which are less then lower bound of the continuous spectrum is
essentially
more simple and has been studied better (see, for example,
~\cite{PAN}).
Analogous (from mathematical point of view) systems were
considered in
fluid mechanics. Namely, the existence of trapped modes above a long
submerged
horisontal circular cylinder of sufficiently small radius, in deep
water,
was proved in ~\cite{U1}. The result was generalized in ~\cite{J}.
Recently,
there has been a revival of interest in predicting those situations in
which trap modes, or acoustic resonances, might occur, because of
their
importance in, for example, the design of turbomashinery ~\cite{PS}.
A series of recent papers has been concerned with both existence
proofs and
numerical algorithms for the computation of trapped modes for
different
geometries ~\cite{EL}, ~\cite{U}. The proofs are based on using of
variational
inequalities. This limitation allows one to look for the effect in
the case
of Neumann boundary condition only. As a result, important case of
Dirichlet
boundary condition has not been considered. But it is Dirichlet
condition that
takes place for quantum waveguide. There is no results about trapped
modes
imbedded in continuous spectrum for quantum waveguides.
We use another approach. To describe complicated problem for system of
resonators coupled with a waveguide through small openings we
construct a
model in which these small apertures are replaced by point-like ones.
The model is based on the theory of self-adjoint extensions of
symmetric
operators. It allows one to obtain the dispersion equation in an
explicit form,
and, hence, to find the eigenvalue in question. It has been proved
earlier
that the model solution is the main term of the asymptotics in the
width
of the aperture of the solution of corresponding realistic problem.
The model
has an advantage in comparison with the conventional approach. In the
framework
of conventional approach the existence of the eigenvalues only can be
proved
and some estimates can be obtained. The model allows one to compute
the
trapped mode and investigate its behaviour. More over, the model
gives us
a possibility to consider both Neumann and Dirichlet boundary
conditions, i.e.
both acoustic and quantum waveguides.
In the second part of the Letter we deal with periodic system of
resonators
connected with a waveguide. It is related with the problem
of nanoelectronics. The discovery of coherent ballistic transport in
narrow channels formed by imposing a potential on a two-dimensional
electron
gas has generated much interest in conduction in small ballistic
structures
~\cite{Been}. Recently Lent et al ~\cite{LL}, ~\cite{PSL} studied
the system which is analogous to our: the quantum channel with a
periodically modulated width.
For our model periodic system the dispersion equation is obtained in
an explicit form and the existence of a band imbedded in the
continuous
spectrum is shown. The model is based on the
theory of self-adjoint extensions of symmetric operators and is
analogous to the well-known zero-range potential method ~\cite{AGHH},
~\cite{Pav}. The operator extension approach allows one to extend the
range of applications of the method ~\cite{JMP1}, ~\cite{JMP2}, and,
particurlarly, to apply it to the problem of quantum transport in
nanostructures ~\cite{PoPo1}, ~\cite{PoPo2}.
Let us describe briefly the structure of the model for the case of one
two-dimensional resonator connected through a small opening with a
waveguide.
Let us consider the Laplace operator in a half plane $\Omega_w$ with
the
Neumann boundary condition. Let $r_0$ be the point of the boundary
common
with the boundary of a bounded domain $\Omega_0$. Restrict the Neumann
Laplacian in $L_2 \left( \Omega_0 \oplus \Omega_w \right)$ to the
set of smooth
functions vanishing at the point $r_0$. The closure of the operator so
obtained is symmetric with the dificiency indices (2, 2). It has
self-adjoint extensions. It is this operator that gives us the model
in
question. We shall consider only one (the most natural) of these
operators.
Its domain consists of the elements of the form
$$u(r)=\left\{ \begin{array}{ll}
a_w G_w (r,r_0, \lambda_0) + u_w (r),
r \in \Omega_w, \\
a_0 G_0 (r, r_0, \lambda_0) +
u_0 (r), r \in \Omega_0,
\end{array}
\right. $$
where $G_0 (r, r_0, \lambda_0)$ be the Green`s function of the Neumann
Laplacian $(-\Delta_0)$ for the domain $\Omega_0$, $G_w(r, r_0,
\lambda)$
be the Green`s function of the
Laplace operator with Neumann boundary condition for the waveguide,
$k^2=\lambda, \quad u_w \in W^2_2(\Omega_w),u_0 \in W^2_2(\Omega_0),
\quad a_w = -a_0, \quad u_w(r_0)=u_0(r_0)$,
$\lambda_0$ be a regular value of the operator. Here $W^2_2$ is
Sobolev
space.
Remark. From a physical point of view this choice of the extension
means
that the condition of "the conservation of the
flux through the aperture" takes place (that is why we call this
extension
"the natural" one). It was shown ~\cite{PopLect}
that one can choose the model parameter $\lambda_0$ in such a way that
the model solution concises with the main term of the asymptotic
expansion
(in $\delta$) of the solution of corresponding "realistic" problem
for domains coupled through small (but non-zero) aperture of the
diameter $\delta$. Namely, for two-dimensional problem we must choose
$\lambda_0 = \delta^{-2}\exp{-2\gamma}, \gamma = 0.5772...$
is Euler constant.
Let there are two identical resonators $\Omega_{0\pm}$ connected
with the waveguide $\Omega_w, \Omega_w = \{ (x,y): \vert d \vert <
d\},$
the boundary of which is represented in Cartesian coordinates $(x,y)$
by two parallel lines $\Gamma_{\pm}$. The points of connections are
$r_{0\pm}$. Let the system is symmetric about the centreline $y = 0$
of
the waveguide. We consider the Laplace operator with Neumann
boundary condition in the space ${\cal H} = L_2 (\Omega_w \oplus
\Omega_{0+} \oplus \Omega_{0-}$. We construct model operator $-\Delta$
using "restriction - extension" procedure described above.
The space can be represented in a form
of orthogonal sum ${\cal H} = {\cal H}_s \oplus {\cal H}_a$, where
${\cal H}_s ( {\cal H}_a)$ is the corresponding subspace of symmetric
(antisymmetric) functions in respect to variable $y$. These subspaces
are
invariant in respect to operator $-\Delta$. Hence,
$$
-\Delta = -\Delta\vert_{{\cal H}_a} \oplus (-\Delta\vert_{{\cal
H}_s}).
$$
Note that functions from the domain of the operator
$-\Delta\vert_{{\cal H}_a}$ satisfy Dirichlet condition on the
centreline
$y = 0$. Consequently, the continuous spectrum are the following
$$
\sigma_c (-\Delta\vert_{{\cal H}_a}) = [\pi^2 (2 d)^{-2}, \infty),
\sigma_c (-\Delta\vert_{{\cal H}_s}) = [0, \infty),
\sigma_c (-\Delta) = [0, \infty).
$$
Using well-known representation for the Green function for Neumann
Laplacian in the waveguide,
$$
G_w (x,y,x*,y*,k)=\sum_{n=0}^{\infty}\frac{\cos(\pi n (y+d) (2d)^{-1})
\cos(\pi n (y*+d) (2d)^{-1})}{2p_n}e^{-p_n\vert x - x* \vert},
$$
$$
p_n(\lambda) = \left\{ \begin{array}{ll}
\sqrt{(\pi n / (2d))^2-k^2}\quad\text{for}\quad
(\pi n / (2d))^2 > k^2,\\
i \sqrt{k^2 - (\pi n /(2d))^2}\qquad\text{for}\quad(\pi n / (2d))^2 >
k^2.
\end{array}
\right.
,$$
one obtains the dispersion equation for the model operator
$-\Delta\vert_{{\cal H}_a}$ in an explicit form:
\begin{equation}\label{1}
\sum^{\infty}_{n,m=1} \frac{1}{(\lambda_{nm} - \lambda)
(\lambda_{nm} - \lambda_0)} =
\sum^{\infty}_{n=1}(\frac{1}{2 p_n(\lambda)} -
\frac{1}{p_n(\lambda_0)}),
\end{equation}
where $\lambda_{nm}$ are eigenvalues for the resonator $\Omega_{0+}$.
Let
\begin{equation}\label{2}
min_{m,n} \lambda_{nm} < \pi^2 / (4 d^2).
\end{equation}
Then for sufficiently great $\lambda_0$
(i.e. for sufficiently small diameter of connection aperture)
the equation (\ref{1}) has real root less then the lower bound of
the continuous spectrum $\pi^2 / (4 d^2)$. Hence, we have an
eigenvalue for the operator $-\Delta\vert_{{\cal H}_a}$ and an
eigenvalue imbedded in the continuous spectrum of the operator
$-\Delta$.
Note that in the framework of conventional approach authors can
prove only that there exists an eigenvalue of the operator
$-\Delta\vert_{{\cal H}_a}$ less then $\pi^2 / (4 d^2)$. Namely,
they show that there exists test function $\psi$ such that
$$
\frac{\int_{\Omega_w \cup \Omega_{0+}\cup\Omega_{0-}}
\vert \nabla \psi \vert ^2 dS}
{\int_{\Omega_w \cup \Omega_{0+}\cup\Omega_{0-}}
\vert \psi \vert ^2 dS} < \pi^2 / (4 d^2).
$$
Hence, in accordance with variational principle there exists an
eigenvalue in question. Thus, they have a proof of existence of an
eigenvalue imbedded in the continuous spectrum but have no effective
algorithm for its determination. Our approach allows one not only to
prove
the existence of such eigenvalues, but also to determine it
approximately using simple procedure. Moreover, the conventional
approach
allows one to consider the Neumann boundary condition only. The
reason is
the following. If one proves that there exist an eigenvalue of the
operator
$-\Delta\vert_{{\cal H}_a}$ less then the lower bound of the
continuous
spectrum of this operator he does not conclude that there is an
eigenvalue imbedded in the continuous spectrum because the continuous
spectrum of the full operator in the case of Dirichlet condition
is not the half axis $[0, \infty)$, but only $[\pi^2 / (4 d^2),
\infty).$
In our model the Dirichlet case may be considered absolutely
parallel to the Neumann one. The model for the Dirichlet boundary
condition is slightly more complicated then the variant described
above.
In this case we had to extend the initial space $L_2$ to construct
the model operator. One can find the description of this variant of
the model
in ~\cite{JMP2}. The model dispersion equation is analogous to
(\ref{1}). The alteration is the following: we must replace the
term $p_n (\lambda)$ by $p_n^D (\lambda), p_n^D (\lambda) =
\sqrt{\lambda - (\pi n / d)^2}.$ The sufficient condition for
existence of eigenvalues imbedded in the continuous spectrum in
the Dirichlet case is
\begin{equation}\label{3}
\pi^2 (2d)^{-2} < min_{m,n} \lambda_{nm} < \pi^2 d^{-2}.
\end{equation}
The dispersion equation allows one to determine this eigenvalue.
One can consider the corresponding periodic system, i.e. when there is
a periodic system (with the period $a$) of identical resonators
$\Omega_{j\pm}, j = 0,\pm 1,\pm 2, ...,$ connected with the waveguide
$\Omega_w$ (note, that analogous periodic system of resonators
connected with a half plain is studied in ~\cite{PPop}).
In this situation we have for the Neumann case the following
representation for an eigenfunction:
$$
\psi(x,k) =
\left\{ \begin{array}{cc}
\alpha_j G_j(r,r_j,k), & r \in \Omega_j,\\
\sum^{\infty}_{j=-\infty} \alpha_j^w G_w(r,r_j,k), & r \in \Omega_w,
\end{array}
\right.
$$
The periodicity of the system leads to Bloch's condition for the
function
$\psi$:
$$
\psi(x+a,k) = \exp{(i p a)} \psi(x,k),
$$
where $p$ is quasimomentum. Hence, one obtains the additional
condition
for the coefficients $\alpha_j^w$:
\begin{equation}\label{4}
\alpha_j^w = \exp{(i p a j)} \alpha_0^w.
\end{equation}
Using the choice of the extension operator described above and
(\ref{4})
one gets the dispersion equation in the form:
$$
\begin{array}{c}
-\left(G_{0+}(r,r_0,\lambda)-G_{0+}(r,r_0,\lambda_0)\right)
\bigg\vert_{r=r_0} =\\
\left(G_w(r,r_0,\lambda)-G_w(r,r_0,\lambda_0)\right)
\bigg\vert_{r=r_0} +
\sum^{\infty}_{j \ne 0, j = -\infty} \exp{(i p a j)}
G_w(r_0,r_j,k).
\end{array}
$$
Using the expression for $G_w$, one gets
$$
\begin{array}{c}
\left(G_{0+}(r,r_0,\lambda)-G_{0+}(r,r_0,\lambda_0)\right)
\bigg\vert_{r=r_0} =\\
\sum_{n=0}^{\infty}(p_n^0 - p_n) (2 p_n p_n^0)^{-1} +
\sum_{n=0}^{\infty} 2 exp{(-p_n a)}\frac{\cos{pa} - \exp{(-p_n a)}}
{1 - 2 \exp{(-p_n a)}\cos{pa} + \exp{(-2 p_n a)}},
\end{array}
$$
where $p_n^0$ is the value of $p_n$ for $\lambda = \lambda_0$.
Using the representation for Green function of the resonator, one
obtains
\begin{equation}\label{3.0}
\begin{array}{c}
\sum^{\infty}_{n,m=1} \frac{1}{(\lambda_{nm} - \lambda)
(\lambda_{nm} - \lambda_0)} = \\
\sum_{n=0}^{\infty}(p_n^0 - p_n) (2 p_n p_n^0)^{-1} +
\sum_{n=0}^{\infty} 2 exp{(-p_n a)}\frac{\cos{pa} - \exp{(-p_n a)}}
{1 - 2 \exp{(-p_n a)}\cos{pa} + \exp{(-2 p_n a)}},
\end{array}
\end{equation}
One can see that if the condition (\ref{2}) takes place, there exist
a band for the operator $-\Delta\vert_{{\cal H}_a}$, i.e. the
band imbedded in the continuous spectrum of the operator $-\Delta$.
One can determine its parameters by solving equation (\ref{3.0}).
The analogous consideration is for the Dirichlet boundary condition.
In this case the sufficient condition for the existence of the
corresponding band is (\ref{3}).
The authors are grateful to Prof. A.G.Kostyuchenko for stimulating
questions and to Prof. L.Parnovski for the discussion.
The work was partly supported by RFBR grants No 96-01-00074 and
No 95-01-00439, Soros Foundation and ANS RF.
\newpage
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\end{document}