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{Perturbation expansions for eigenvalues and eigenvectors for a
rectangular membrane subject to a restorative force}

\author{Joyce R. McLaughlin}
\address{Rensselaer Polytechnic Institute, Troy, NY 12180}

\author{Arturo Portnoy}
\address{Rensselaer Polytechnic Institute, Troy, NY 12180}

\commby{Michael Taylor}

\date{May 16, 1997}

\subjclass{Primary 35P20} 

\keywords{Perturbation expansion, eigenvalue, eigenvector, membrane,
inverse nodal problem}

\dateposted{August 19, 1997}
\PII{S 1079-6762(97)00027-9}

\copyrightinfo{1997}{American Mathematical Society}


Series expansions are obtained for a rich subset of eigenvalues and
eigenfunctions of an operator that arises in the study of rectangular
membranes: the operator is the 2-D Laplacian with restorative force term
and Dirichlet boundary conditions. Expansions are extracted by considering
the restorative force term as a linear perturbation of the Laplacian;
errors of truncation for these expansions are estimated. The criteria
defining the subset of eigenvalues and eigenfunctions that can be studied
depend only on the size and linearity of the perturbation. The results are
valid for almost all rectangular domains.



In this summary we consider a rectangular, homogeneous membrane and a force
acting on it that depends linearly on the displacement from equilibrium.
The eigenvalues and eigenfunctions are well known when the restorative
force term is zero, prompting us to treat the case of nontrivial
restorative force as a linear perturbation of the homogeneous case.

The distance between unperturbed eigenvalues is very important in studying
how perturbed eigenvalues and eigenfunctions behave. It turns out that even
in the cases where all unperturbed eigenvalues are distinct, sufficiently
large eigenvalues can get arbitrarily close together. These arbitrarily
small differences appear as divisors in the expansions, making it difficult
to prove the expansions worthy.

This paper is inspired by the work of McLaughlin and Hald, \cite{hm:brief}
and \cite{hm:whole}. They derive asymptotic expansions for almost all
eigenvalue, eigenfunction pairs in order to solve the inverse nodal
problem, with corresponding error estimates given in several norms. They
estimate very sharp error bounds for specific truncations of the
perturbation expansion and show that their expansions hold for almost all
eigenvalues and eigenfunctions. The criteria defining this set of ``good''
eigenvalue, eigenfunction pairs are not completely known a priori.
They present three a priori conditions and a fourth condition that depends
on the values of the perturbed eigenvalues. We present the form and error
bounds of arbitrary truncations, and we do that for a well-defined (defined
a priori) subset consisting of almost all eigenvalues. Also, our
restrictions on the parameters, including the smoothness of $q$, are less
stringent. We estimate cruder bounds on the error of truncation. We do not
give details here, but in \cite{mp:whole} we show that if the additional
restrictions on smoothness and other parameters that are imposed in
\cite{hm:whole} are made, then their sharper estimates are valid for our
collection of ``good'' eigenvalue, eigenfunction pairs. Here we provide a
rich subset of eigenvalue, eigenfunction pairs, determined exclusively with 
a priori lattice point requirements, that can be used to obtain our
error bounds, to obtain the error bounds in \cite{hm:whole} and to solve
the inverse nodal problem defined in \cite{hm:whole}.

The strategy used to find these expansions follows:
\item We build on three key results from \cite{hm:whole}: The first tells
us that for almost all rectangular membranes, almost all eigenvalues are
well separated; the second tells us that eigenvalues with close neighbors
rarely have corresponding eigenfunctions with similar oscillation
properties; finally, the third condition eliminates points in the lattice
that are too close to either axis. All three results define ``thin''
subsets of ``bad'' lattice points with a priori criteria.
Disregarding these subsets is important but insufficient to justify the
expansions we give here.
\item We add an additional a priori condition which strips another
{\em thin} subset of ``bad'' eigenvalues, allowing us to prove the
expansions convergent for the remaining large set of eigenvalues and
eigenfunctions. Geometric considerations are used to prove that this new
set is ``thin''. This condition is the main effort of this work. It
replaces the fourth criterion imposed in \cite{hm:whole}, p. 68.
\item Explicit $L^{\infty }$ bounds are given for the errors of truncation
of the eigenfunction and eigenvalue expansions.

Four other papers are listed in the Bibliography that deal with the
solution of problems similar to ours: \cite{fkt}, \cite{fried}, \cite{karp}
and \cite{karp2}. We point out differences, similarities, advantages and
disadvantages between our approach and theirs in what follows.

First, we should define the problem these three papers examine: the
differential operator is the same, but the boundary conditions are
different. All three consider an unbounded domain, but assume $q$ to be
periodic in each variable. This turns out to be equivalent to solving a set
of problems restricted to a finite domain with periodic and quasi-periodic
boundary conditions.

In \cite{fkt}, Feldman, Kn\"{o}rrer and Trubowitz show that a rich set of
eigenvalue, eigenfunction pairs of Schr\"{o}dinger's equation with periodic
potential are {\em perturbatively stable}, which means that the difference
between perturbed and unperturbed eigenvalues and eigenfunctions is
asymptotically small, measured in inverse fractional powers of the
eigenvalue; the error bound is given only for the first truncation of the
expansions. They accomplish their result by disregarding small subsets of
{\em bad} lattice points and their corresponding eigenvalue-eigenfunction
pairs. In fact, this discarded set is defined by three conditions that
involve geometric properties of the lattice. We will also discard a thin
subset of {\em bad} lattice points. Our discarded set is defined by four
conditions, two equivalent to theirs and two that are of a different nature
but also depend only on geometric properties of the lattice points.

In \cite{fried}, Friedlander gives another proof for results similar to
those in \cite{fkt}. To do this, conditions equivalent to our first two are
also imposed, plus an additional condition which uses information not known
a priori, namely, the perturbed eigenvalues.

Both these papers are concerned only with the first truncation of the
perturbation expansions, and both exhibit good estimates on the difference
between perturbed and unperturbed eigenvalues. It should be noted that in
both these papers, the bound for the difference between perturbed and
unperturbed eigenfunctions is not an $L^{\infty}$ bound. Having an
$L^{\infty}$ bound is necessary to use these estimates to solve the inverse
nodal problem.

In \cite{karp}, Karpeshina exhibits the complete perturbation expansions
for a rich set of eigenvalues and eigenprojectors for Schr\"{o}dinger's
equation with periodic potential with the additional assumption that the
potential is a finite trigonometric polynomial in the spatial variables;
she outlines how the procedure could be generalized for a smooth potential.
Error estimates for truncations of these expansions are presented, and
conditions defining subsets of {\em bad} eigenvalues are defined as well.
The same pair of conditions common to all papers are represented here as
well. The third condition in \cite{karp} is similar in nature, but
different, to ours.

In \cite{karp2}, Karpeshina presents the complete perturbation expansions
for a rich set of eigenvalues and eigenprojectors for Schr\"{o}dinger's
equation with periodic potential. In this paper, assumptions on the
smoothness of the potential are relaxed considerably. Estimates for errors
of truncation are presented for a trace class norm, which, in fact, makes
these estimates valid in $L^{\infty}$. The only drawback of the technique
is that although the results are proved for a measure one subset of
quasimomenta, a particular instance of quasimomenta for which the results
are valid is not produced.

In what follows we present the statement of the key result in a
setting that is not as general as possible, but makes ideas more
accessible to the reader.

\section*{Main result}

We want to study the behavior of the eigenvalues and eigenfunctions satisfying
-\Delta u + qu = \lambda u      \label{eq:basic1}
on the rectangular domain $R = [0,\pi/a]\times [0,\pi]$ ($a>1$), with
Dirichlet boundary conditions. Note that $q$ is in general not constant.

In order to better understand the effect of the $qu$, or {\em restorative
force} term, we have to refer to the original equation from which this
eigenvalue problem was derived, that is, the wave equation
$$ v_{tt} + gv = \rho ^2\Delta v $$
with Dirichlet boundary conditions and two prescribed initial conditions,
where $\rho ^2$ is the quotient of the constant tension over the constant
density of the membrane. The term $gv$ is called restorative when $g\geq 0$
because it has the effect of wanting to return the system to a state of
equilibrium or zero displacement. A way of thinking about the $gv$ term is
to set $\rho ^2\Delta v = 0$ and study $v_{tt} + gv = 0$. This is the
equation governing the motion of a linear spring, where $g$ is a measure of
the stiffness of springs attached to the membrane at each point in our

To obtain the eigenvalue problem we must consider solutions of the form $v
= e^{i\sigma t} u$, where $u$ is time independent. Plugging in and
$$-\sigma ^2u + gu =  \rho ^2\Delta u. \nonumber$$
Slightly rearranging the terms we can see that $q = g/\rho ^2\geq 0$ and
$\lambda = \sigma ^2/\rho ^2$.

The eigenvalues and normalized eigenfunctions of (\ref{eq:basic1}) when $q
\equiv  0$ are well known for rectangular domains:
\lambda_{\alpha } & = & (an)^2 + m^2 \; = \; |\alpha |^2, \nonumber \\
u_{\alpha } & = &
\sin( %\it
\sin( %\it
\textit{my}), \nonumber
where $\alpha \in L$ and our index set $L=\{(an,m)|n,m=1,2,3,\dots\}$ is an
integer lattice.
We will treat the case of $q \not=  0$ as a {\em perturbation} of the
simpler, well-understood case ($q \equiv  0$). So our new formulation for
the problem will be
-\Delta u(\epsilon )  + \epsilon qu(\epsilon ) = \lambda (\epsilon )
u(\epsilon ). \label{eq:basic2}
We use $\epsilon $ as a book-keeping parameter in the end set $\epsilon =
1$, thus solving the original problem.

  Let $J=(1,a_0)$ be given. Define
 $V = \Bigl\{ a\in J \Bigr|$ there exist $0 < \delta < \epsilon_0/6$ and
$K > 0$ such that for all $p,q>0: \bigl|a^2 -
\frac{p}{q}\bigr|>K/q^{2+\delta} \Bigr\}.$
Let $q$ be such that $$\int_{R}q\,dx = 0,$$ and $$|q|_l =
\left\{\sum_{\alpha \in L}|(q,v_{\alpha})|^2|\alpha
|^{2l}\right\}^{1/2}<\infty,$$ with $l \geq 2$. Then there exists a finite,
positive number $D_1$ and an exceptional set of lattice points
$\breve{M}(a) \subset L(a)$ depending on $a$ such that

a) $meas(J\backslash V)=0$ and $\breve{M}(a)$ has density 0 in $L(a)$ for
all $a \in V$ in the sense that
$$\lim_{r\rightarrow \infty}\frac{\#\breve{M}(a)\cap B(0,r)}{\#L(a)\cap

b) For every $\alpha \in L(a) \backslash (\breve{M}(a)\cup B(0,D_1))$, there
exists a circular contour $\Gamma _{\alpha }$ in the complex plane,
centered at $\lambda_{\alpha }$, containing one and only one eigenvalue
throughout the perturbation. Moreover, there is a unique eigenvalue of the
variable coefficient problem {\rm (\ref{eq:basic1})}, which we denote by
$\lambda _{\alpha q}$, satisfying
|\lambda_{\alpha q} - \lambda_{\alpha }| &\leq& \left(2 +
30\sqrt{a}|q|\right)|q|_l^2|\alpha |^{-1/2 + 2\delta }.\nonumber

This is the bound for the error made by the 1st truncation of the
perturbation expansion for the eigenvalue. In general, the perturbation
expansion for the eigenvalue is given by
\lambda _{\alpha q} & = &\lambda _{\alpha } + \sum^{\infty }_{n=0} (-1)^n
\sum_{\beta _0 \in L}\cdots\sum_{\beta _{n} \in L}\left\{\frac{1}{2\pi
i}\int_{\Gamma _{\alpha }} \frac{\left(\lambda _{\beta_0 }  - \lambda
_{\alpha }\right)(u_{\beta _0},qu_{\beta _1})}{(\lambda _{\beta _0} -
\xi)^2 }\right.\nonumber\\
& &\times\left.\frac{(u_{\beta _1},qu_{\beta _2})}{\lambda _{\beta _1} - \xi }
\frac{(u_{\beta _{n}},qu_{\beta _0})}{\lambda _{\beta _{n}} - \xi
}\,d\xi\right.\nonumber \\
&   &  -\;\left.\frac{1}{2\pi i}\int_{\Gamma _{\alpha }} \frac{(u_{\beta
_n},qu_{\beta _0})}{\lambda _{\beta _0} - \xi } \frac{(u_{\beta
_0},qu_{\beta _1})}{\lambda _{\beta _1} - \xi }\cdots\frac{(u_{\beta
_{n-1}},qu_{\beta _n})}{\lambda _{\beta _n} - \xi }\,d\xi\right\},\nonumber
and a bound for the error made by the $m$th truncation is
$$ \sqrt{a}|q|12(m+1)|q|_l^{m-1}|\alpha |^{(-1/4 + \delta)(m-1)}. $$

A suitable multiple of the corresponding eigenfunction satisfies
$$\|u_{\alpha q} - u_{\alpha }\|_{L^{\infty}} \leq 3\sqrt{a}|q|_l^2|\alpha
|^{-1/2 + 2\delta }.$$

This is a bound for the error made by the 1st truncation of the
perturbation expansion for the eigenfunction. In general, the perturbation
expansion of the eigenfunction is given by
u_{\alpha q} & = & u_{\alpha }
+ \sum^{\infty }_{n=1} (-1)^{n+1}\sum_{\beta _0 \in L} \cdots\sum_{\beta
_{n-1} \in L} \frac{1}{2\pi i}\int_{\Gamma _{\alpha }} \frac{(u_{\beta
_0},qu_{\beta _1})}{\lambda _{\beta _0} - \xi }\times\cdots
& &\times\frac{(u_{\beta
_{n-2}},qu_{\beta _{n-1}})}{\lambda _{\beta _{n-2}} - \xi }\frac{(u_{\beta
_{n-1}},qu_{\alpha })}{\lambda _{\beta _{n-1}} - \xi }\frac{1}{\lambda
_{\alpha} - \xi }u_{\beta_ 0}\,d\xi \nonumber
and a bound for the $L^{\infty}$ error made by the $m$th truncation is
$$2\sqrt{a}|q|_l^{m}|\alpha |^{(-1/4 + \delta )m},$$
where $0 < \delta < 1/10$ can be made arbitrarily small.

%{\bf Remark:} 
The reader will observe that the bound given for the error
made by the 1st truncation for both the eigenvalue and eigenfunction is
stronger than that implied by the error for the 1st truncation of the
series expansion. To obtain the better bound we more carefully bound the
$n=1$ and $n=2$ terms of the corresponding series expansion.
Because our main contribution involves the definition of the exceptional
subset $\breve{M}(a) \subset L(a)$, we present its defining elements in
what follows:
$M_{10}(a) = \Bigl\{\alpha \in L \Bigl|$ there exists $\beta \in L$, $\beta
\not= \alpha$, and $\Bigl||\alpha|^2 - |\beta|^2\Bigr| <
C_0|\alpha|^{-\epsilon _0}\Bigr\}$,
$M_{11}(a) = \Bigl\{\alpha \in L\Bigl|$ there exists $\beta \in L$, $\beta
\not= \alpha$, and $|\alpha - \beta| < C_1|\alpha |^{\epsilon_1}$,
$|\lambda_{\alpha 0} - \lambda_{\beta 0}| <
$M_{12}(a) = \Bigl\{\alpha=(an,m) \in L\: \Bigl|$ we have that $m<(\delta
C_1)^{1/(1-p)}(an)^p$  or  $an<(\delta C_1)^{1/(1-p)}m^p\Bigr\}$,
M_{13}(a) = \Bigl\{\alpha \in L\backslash M_{10}\: &\Bigl|&\: \exists\ 
\beta ,\gamma \in L,\ \beta ,\gamma \not= \alpha, \nonumber \\
&   &\: 0<|\beta - \gamma |
\max %}
\left\{(1-\epsilon _2)/p,1+(3\epsilon _0)/(2\epsilon
_1),1+(\epsilon _3+\epsilon _0)/\epsilon _1\right\},$$
$$0 < \epsilon_0 < (\epsilon _2 - \epsilon _1)/2, \hspace{0.5 in} 0 <
\epsilon _3 <   \epsilon _2 - \epsilon _1 - 2\epsilon_0,$$
$$C_0 > C_3,$$
$$2(C_3/C_0)\left(1/\sqrt{1-C_3/C_0}\right)^{\epsilon_3 + \epsilon_0}