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\title{On the point and continuous spectra for
coupled quantum waveguides and resonators}
\author{I.Yu.Popov\\
Department of Higher Mathematics,\\
Leningrad Institute of Fine Mechanics and Optics,\\
14 Sablinskaya, 197101 Leningrad, Russia, USSR.\\}
\begin{document}
\maketitle
\begin{abstract}
A system of quantum resonators connected through small apertures
with a quantum waveguide and a system of waveguides coupled through
small
windows are considered in the case of Dirichlet boundary condition.
Solvable model based on the operator extension theory in
Pontryagin space is suggested for the description of trapped
modes imbedded in the continuous spectrum for the systems.
The model allows one not only to prove the existence of such modes
but also to suggest an effective and simple algorithm for its
determination. A system of quantum waveguides coupled laterally
through small windows is considered. The existence of eigenvalue
imbedded in the continuous spectrum is shown. The case of
periodic set of windows 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
\section{Introduction}
The problem of eigenvalues imbedded in the continuous spectrum is
studied now actively both from mathematical and physical points of
view.
Mathematicians are trying to reveal general features of the
phenomenon by studying of concrete problems (see, for example,
\cite{NP}). The problem is important for physicists because it 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 describe systems of
quantum resonators connected with a waveguide and waveguides coupled
through small windows. 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 \cite{U1}, \cite{J}, \cite{PS}, \cite{EL}, \cite{U}.
The conventional proofs of existence of eigenvalues imbedded in the
continuous spectrum are based on using of variational
inequalities. This limitation allows one to look for the effect in
the case
of the Neumann boundary condition only. As a result,
important case of the Dirichlet
boundary condition has not been considered. But it is the
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.
To construct the model for the Dirichlet boundary condition we
have to work in the space with indefinite metric (see \cite{JMP2}).
The model is analogous to well-known zero-range potential approach
\cite{AGHH}, \cite{Pav}.
In the first section we describe briefly the model.
In the second part of the Letter we deal with system of resonators
connected with a waveguide. It is related with the problem
of nanoelectronics, particurlarly with the investigation of
ballistic transport in narrow channels ~\cite{Been}.
The extension theory model for more simple case of Neumann
boundary condition for resonators connected with a waveguide was
constructed in \cite{PoPo2}, \cite{PoPo3}.
The third section is devoted to waveguides coupled through small
windows (see, for example, \cite{K}). The existence of
eigenvalues which are less then the continuous
spectrum for this problem was shown in \cite{EV}. In our model
this fact is rather simple. We study the situation when
eigenvalues are imbedded in the continuous spectrum. The model allows
us to investigate more complicated case of periodic system of
windows (see \cite{TP} for corresponding example in nanoelectronics)
and to show that there exist a band imbedded in the continuous
spectrum.
It may be mentioned that the problem of eigenvalues imbedded
in the band spectrum for two-dimensional periodic array of quantum
dots in constant magnetic field which is orthogonal to the system
plane is considered in \cite{GP1}.
\section{Description of the model}
Describe briefly the model for the case of using an
indefinite metric space. Let $\Delta^{1,2}$ be the Laplace operators
with the Dirichlet boundary conditions in the domains
$\Omega^{1,2}$ with smoooth boundaries having common point
$r_0$. Conventional "restriction- extension"
procedure for constructing the model is the following.
One restricts the initial self-adjoint operator on the set of
functions vanishing near the point $r_0$. The obtained symmetric
operator has self-adjoint extensions which gives us the model in
question \cite{JMP2}. But in the case of the Dirichlet boundary
condition the restriction of the operator on the set of smooth
functions vanishing near the point $r_0$ is essentially
self-adjoint operator, i.e. the procedure fails.
To construct the model we have to use
extended space with indefinite metric \cite{JMP2}. Consider
how to take into account the dipole moment. It is necessary to
add the function
$h_{-1}^{1,2} = \frac{\partial}{\partial n_y}
G^{1,2}(r,r',k_0)\vert_{r'=r_0}$
to the initial space $L_2$. Here $G^{1,2}$ is Green function
for the Dirichlet problem in $\Omega^{1,2}$. Let $A^{1,2}$ be
the following set of functions:
$$
A^{1,2} = \left\{ f(r): f \in L_2(\Omega^{1,2}),
\int_{\Omega^{1,2}} f(r)\vert r - r_0 \vert ^{-2} d r -
\text{converges}\right\}.
$$
Note that these functions have roots of corresponding
multiplicity at $r_0$.
Let us introduce the following chain of functions:
$$
h_{-1}^{1,2},
h_1^{1,2} = (-\Delta^{1,2} - \lambda_0)^{-1} h_{-1}^{1,2}.
$$
Here $k_0$ is some imaginary number ($k_0 = \lambda_0$ is a
regular point of the operator $-\Delta^{1,2}$). Let us consider
the set $\tilde A^{1,2}:$
$$
\tilde A^{1,2} = \left\{ \tilde f^{1,2}:
\tilde f^{1,2} = f^{1,2} + C_1^{1,2} h_1^{1,2} + C_{-1}^{1,2}h_{-
1}^{1,2}
\right\} .
$$
Here $f^{1,2} \in A^{1,2}.$ We define a scalar product in
$\tilde A^{1,2}$ by the following manner:
$$
\begin{array}{c}
(\tilde f, \tilde g)_{\tilde A^{1,2}} =
(f,g)_{L_2} + \int_{\Omega^{1,2}}f(r)
(\overline{C_1^g h_1^{1,2} + C_{-1}^g h_{-1}^{1,2}}) d r + \\
\int_{\Omega^{1,2}}\overline{g(r)}
(C_1^f h_1^{1,2} + C_{-1}^f h_{-1}^{1,2}) d r +
C_1^f\overline{C_1^g}(h_1^{1,2},h_1^{1,2})_{L_2} +\\
(C_1^f\overline{C_{-1}^g} + C_{-1}^f\overline{C_{1}^g})
(h_{-1}^{1,2},h_1^{1,2})_{L_2}.
\end{array}
$$
The set $\tilde A^{1,2}$ is imbedded into Pontryagin space $\Pi_1$
by conventional way \cite{Sh}.
Let $\tilde \Delta^1$ be the Laplace operator with the domain
$$
D(\tilde \Delta^1) = \left\{ \tilde f :
\tilde f \in \tilde A^1, f \in W_2^{2,loc}(\Omega^1),
f = f_1 + C h_1^1 \right\}.
$$
Here $f_1, (-\tilde \Delta^1 - \lambda_0)f_1 \in A^1.$ On the set
$A^1$ the operator $\tilde \Delta^1$ acts as the Laplace operator
and on the chain $h_p^1}$ the operator $(-\tilde \Delta^1 -
\lambda_0)$
is a shift operator: $(-\tilde \Delta^1 - \lambda_0)h^1_1 = h_{-1}^1.$
The operator $-\tilde \Delta^1$ is self-adjoint one.
Restrict this operator onto the following set:
$$
D(\Delta^1_0) = \left\{ f: f \in D(\tilde \Delta^1),
((-\tilde \Delta^1 - \lambda_0)f, h_{-1}^1) = 0 \right\}.
$$
The obtained operator $\Delta^1_0$ is a symmetric one and has
deficiency indices $(1,1)$. The construction for $\Omega^2$ is
analogous. Then the operator $\Delta_0 =\Delta^1_0 \oplus \Delta^2_0$
is symmetric and has deficiency indices $(2,2)$. It has
self-adjoint extensions. Note that the domain of the adjoint
operator is the following:
$$
D(\tilde \Delta^*_0) = \left\{ \tilde f :
\tilde f \in \tilde A^1, f = f_1 + C h_1^1 \right\}.
$$
Here $f_1, (-\tilde \Delta^1 - \lambda_0)f_1 \in A^1.$
To describe the domain of the extensions
it is necessary to find a linear set of elements from
$D(\Delta^*_0)$ which satisfy the condition of the annihilation
of the following boundary form:
$$
J(f,g) = ((-\tilde \Delta_0^* - \lambda_0)\tilde f,\tilde g) -
(\tilde f, (-\tilde \Delta_0^* - \lambda_0)\tilde g).
$$
It easy to show that $D(\Delta_0)$ consists of all elements from
$D(\tilde\Delta)$ which satisfy the condition $C^1_1 = C^2_1 = 0.$
Using this fact and known asymptotics of the Green function near the
point of the boundary, one obtains after brief calculation:
$$
J(f,g) = C_1^{f,1}\overline{C_{-1}^{g,1}} -
C_{-1}^{f,1}\overline{C_{1}^{g,1}} +
C_1^{f,2}\overline{C_{-1}^{g,2}} -
C_{-1}^{f,2}\overline{C_{1}^{g,2}}.
$$
Using conventional procedure of construction of Lagrange planes from
simplectic geometry, we come to the following theorem.
{\bf Theorem 1}. {\it The set of self-adjoint extensions of the
operator} $-\Delta_0$ {\it consists of the sets of operators}
$\left\{ -\Delta_A \right\}, \left\{ -\Delta_B \right\}$,
{\it the domains of which consist of all elements from}
$D(-\Delta_0^*$ {\it satisfying the conditions}
$$
\left( \begin{array}{c}
C_1^1 \\ C_1^2 \end{array} \right)
= \left( \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array} \right)
\left( \begin{array}{c}
C_{-1}^1 \\ C_{-1}^2 \end{array} \right)
$$
{\it or}
$$
\left( \begin{array}{c}
C_{-1}^1 \\ -C_1^2 \end{array} \right)
= \left( \begin{array}{cc}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{array} \right)
\left( \begin{array}{c}
C_1^1 \\ C_{-1}^2 \end{array} \right),
$$
{\it where} $A, B$ {\it are Hermitian matrices.}
\section{Waveguide with coupled resonators}
Let there are two identical resonators $\Omega_{0\pm}$ connected
with the waveguide $\Omega_w, \Omega_w = \{ (x,y): \vert y \vert <
d\},$
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 Dirichlet
boundary condition in the space $L_2 (\Omega_w \oplus
\Omega_{0+} \oplus \Omega_{0-})$. We construct model operator $-
\Delta$
using "restriction - extension" procedure described above, i.e.
using the extension of the initial space.
The extended space ${\cal H}$ 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 the 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
$$
\sigma_c (-\Delta\vert_{{\cal H}_a}) = [\pi^2 d^{-2}, \infty),
\sigma_c (-\Delta\vert_{{\cal H}_s}) = [\pi^2 (2 d)^{-2}, \infty),
\sigma_c (-\Delta) = [\pi^2 (2 d)^{-2}, \infty).
$$
In accordance with theorem 1, there is a family of model
operators, but we shall deal with one fixed extension which
corresponds to the following matrix $B: b_{11} = b_{22} = 0,
b_{12} = b_{21} = -1.$ It was shown \cite{KPop} that this choice
of the extension gives us correct correlation between the model
and "realistic" problems.
The following representation for the Green function for
Dirichlet Laplacian in the waveguide is well-known:
$$
G_w (x,y,x^*,y^*,k)=\sum_{n=0}^{\infty}\frac{\sin(\pi n (y+d) (2d)^{-
1})
\sin(\pi n (y^*+d) (2d)^{-1})}{2p_n}e^{-p_n\vert x - x^* \vert},
$$
$$
p_n(\lambda) = \left\{ \begin{array}{l}
\sqrt{(\pi n / (2d))^2-k^2}\quad\text{for}\quad
(\pi n / (2d))^2 > k^2,\\
i \sqrt{k^2 - (\pi n /(2d))^2}\qquad\text{for}\quad(\pi n / (2d))^2 <
k^2.
\end{array}
\right.
$$
Using this expression and analogous one for the Green function
for the resonator $\Omega_{\pm}$ and taking into account "the
boundary condition at the points $r_{0\pm}$" for chosen extension
one comes to the following dispersion equation for the model operator
$-\Delta\vert_{{\cal H}_a}$:
\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 $\Omega_{0+}$.
{\bf Theorem 2.} {\it If}
\begin{equation}\label{2}
\pi^2 / (4 d^2) < min_{m,n} \lambda_{nm} < \pi^2 / (d^2).
\end{equation}
{\it then for sufficiently great} $\vert \lambda_0 \vert$
{\it the model operator} $-\Delta$ {\it has an eigenvalue
imbedded in the continuous spectrum.}
Really, condition (\ref{2}) leads to the fact that
equation (\ref{1}) has real root less then the lower bound of
the continuous spectrum $\pi^2 / (d^2)$. Hence, we have an
eigenvalue for the operator $-\Delta\vert_{{\cal H}_a}$
If $\lambda_0$ tends to infinity (i.e. for sufficiently small
diameter of connection aperture (see \cite{KPop})) then the root
tends to $\pi^2 / (d^2)$. Hence, for sufficiently great
$\lambda_0$ it becames greater then $\pi^2 / (2 d)^2$.
Consequently, we obtain an
eigenvalue imbedded in the continuous spectrum for the operator
$-\Delta$.
Note that in the framework of conventional approach authors
deal with Neumann Laplacian and can
prove only that there exists an eigenvalue of the operator
$-\Delta\vert_{{\cal H}_a}$ less then $\pi^2 / (d^2)$. 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.
In such a way, they get a proof of existence of an
eigenvalue imbedded in the continuous spectrum for Neumann Laplacian
which has the continuous spectrum is $[0,\infty)$ (but they have
no effective algorithm for its determination).
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 $[\pi^2 / (d^2),
\infty).$
Our approach allows one not only to prove
the existence of such eigenvalues, but also to determine it
approximately using simple procedure. One can see that our approach
allows one to consider both Dirichlet and Neumann cases (for
Neumann case the model is more simple (see \cite{PoPo3})
because there is not necessity to extend the initial space $L_2$).
One can consider the corresponding periodic system of coupled
resonators but we shall deal with another situation.
\section{Waveguides coupled through windows}
Let us consider two identical waveguides $\Omega_{w\pm}$
coupled through point-like window at the point $r_0$. Using the
model described above for Dirichlet Laplacian for the system we
come to the following dispersion equation:
\begin{equation}\label{3}
\sum^{\infty}_{n=1}(\frac{1}{p_n(\lambda)} -
\frac{1}{p_n(\lambda_0)}) = 0,
\end{equation}
If $\lambda < \lambda_1$ ($\lambda_1$ is the minimal eigenvalue for
the cross-section) then there exist an eigenvalue $\lambda_*$
less then $\lambda_1$. It tends to the lower bound of the continuous
spectrum when $\lambda_0 \rightarrow \infty$. Note that corresponding
result for "realistic" problem (when there is a window of finite
width) is in \cite{EV}. One can use the comparison between the model
and "realistic" results to ensure an appropriate correlation
between model and "realistic" solutions by choosing the model
parameter $\lambda_0$. Note that the condition
$\lambda_0 \rightarrow \infty$ corresponds to the fact that the
width of the window in the corresponding "realistic" problem
tends to zero. It is shown in \cite{EV} that there are several
eigenvalues but if the width of the window tends to zero, there
is only one eigenvalue. In our model we have one eigenvalue which
is less then the lower bound of the continuous spectrum. But we
can modify the model by using so-called "zero-range potentials
with internal structure" (see, for example, \cite{GP}) to obtain
several eigenvalues.
Consider now three waveguides $\Omega_{1,2,3}$ coupled through
point-like windows (fig. 1a). We construct our model for the
Dirichlet Laplacian and represent the model operator in the form
$$
-\Delta = -\Delta\vert_{{\cal H}_a} \oplus (-\Delta\vert_{{\cal H}_s})
$$
as above. Note that to construct the model for the operator
$-\Delta\vert_{{\cal H}_a}$ is to construct
the model for the Dirichlet Laplacian for the system shown
on figure 1b. Hence, one gets for the operator
$-\Delta\vert_{{\cal H}_a}$ the following dispersion equation:
\begin{equation}\label{4}
\sum^{\infty}_{n=1}(\frac{1}{p^1_n(\lambda)} -
\frac{1}{p^1_n(\lambda_0)}) +
\sum^{\infty}_{n=1}(\frac{1}{\tilde p^2_n(\lambda)} -
\frac{1}{\tilde p^2_n(\lambda_0)}) = 0,
\end{equation}
where
$$
p^1_n(\lambda) = \left\{ \begin{array}{l}
\sqrt{(\pi n / (2d_1))^2-k^2}\quad\text{for}\quad
(\pi n / (2d_1))^2 > k^2,\\
i \sqrt{k^2 - (\pi n /(2d_1))^2}\qquad\text{for}\quad
(\pi n / (2d_1))^2 < k^2,
\end{array}
\right.
$$
$$
\tilde p^2_n(\lambda) = \left\{
\begin{array}{l}
\sqrt{(\pi n / (d_2))^2-k^2}\quad\text{for}\quad
(\pi n / (d_2))^2 > k^2,\\
i \sqrt{k^2 - (\pi n /(d_2))^2}\qquad\text{for}\quad
(\pi n / (d_2))^2 < k^2.
\end{array}
\right.
$$
Let $d_1 < d_2 \leq 2d_1$.
If $\lambda < \pi^2 / (2 d_1)^2$ then there is a root of
equation (\ref{4}), i.e. an eigenvalue of the operator
$-\Delta\vert_{{\cal H}_a}$. For sufficiently large
$\vert\lambda_0\vert$
it is an eigenvalue imbedded in the continuous spectrum
$[(\pi n / (2 d_2))^2,\infty)$ for the full operator $-\Delta$.
Let us consider a situation when there is
a periodic system (with the period $a$) of point-like windows at
the points $r_{j\pm}, j = 0,\pm 1,\pm 2, ...,$ (fig. 1c)
(note, that the model for Neumann Laplacian for periodic
system of resonators
connected with a half plane is studied in ~\cite{PPop}).
Let for simplicity $d_2 = 2 d_1$.
In this situation we have for the Dirichlet case the following
representation for an eigenfunction of the model operator
$-\Delta\vert_{{\cal H}_a}$:
$$
\psi(x,k) =
\left\{ \begin{array}{cc}
\sum^{\infty}_{j=-\infty} \alpha_j^1
\frac{\partial G_1}{\partial n}(r,r_{j+},k), & r \in \Omega_1,\\
\sum^{\infty}_{j=-\infty} \alpha_j^w
\frac{\partial G_w}{\partial n}, & r \in \Omega_w,
\end{array}
\right.
$$
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 $\alpha_j^w (\alpha_j^1)$:
\begin{equation}\label{5}
\alpha_j^w = \exp{(i p a j)} \alpha_0^w.
\end{equation}
Using the choice of the extension operator described above,
condition $d_2 = 2 d_1$ and (\ref{5})
one gets the dispersion equation in the form:
$$
\left(G_w(r,r_0,\lambda)-G_w(r,r_0,\lambda_0)\right)
\bigg\vert_{r=r_0} +
\sum^{\infty}_{j \ne 0, j = -\infty} \exp{(i p a j)}
G_w(r_0,r_j,k) = 0.
$$
Using the expression for $G_w$, one gets
\begin{equation}\label{6}
\sum_{n=0}^{\infty}(p_n^1(\lambda_0) - p_n^1(\lambda)
(2 p_n^1(\lambda) p_n^1(\lambda_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{equation}
One can see that if $\lambda < \pi^2 / (2 d_1)^2$ then there is 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{6}).
The analogous consideration is for the Neumann boundary condition.
\section{Conclusion}
A problem of eigenvalues and bands imbedded in the continuous
spectrum is considered. We deals with a system of quantum
resonators connected through small orifices with a quantum waveguide
and waveguides coupled through small windows. The approach is based
on the theory of self-adjoint extensions of symmetric operators.
Note that conventional approaches (see Introduction) allows one to
use variational methods only. As a result, the case of the Neumann
boundary condition only has been considered in previous works and
the existence only of the eigenvalues imbedded in the continuous
spectrum has been proved. The advantage of our approach is that
it allows one to obtain asymptotics of the corresponding solutions
(and, consequently, the asymptotics for the eigenvalues) instead
of variational inequalities for the eigenvalues. Hence, we have
an effective instrument to compute eigenvalues and bands
imbedded in the continuous spectrum. The suggested approach may
be used for the investigation of other spectral problems.
{\bf Acknowledgements}
The author is grateful to Prof. A.G.Kostyuchenko for stimulating
questions, to Prof. L.Parnovski, Prof. P.Exner and Prof. S.Vugalter
for the discussion. The work was partly supported by RFBR grants
No 96-01-00074 and No 95-01-00439, Soros Foundation, and EC-Russia
Collaboration Contract ESPRIT P9282 ACTCS.
\newpage
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\end{thebibliography}
\end{document}