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## Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2014

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We consider the eigenvalue problem for the Reissner-Mindlin system arising in
the study of the free vibration modes of an elastic clamped plate. We provide
quantitative estimates for the variation of the eigenvalues upon variation of
the shape of the plate. We also prove analyticity results and establish
Hadamard-type formulas. Finally, we address the problem of minimization of the
eigenvalues in the case of isovolumetric domain perturbations. In the spirit of
the Rayleigh conjecture for the biharmonic operator, we prove that balls are
critical points with volume constraint for all simple eigenvalues and the
elementary symmetric functions of multiple eigenvalues.; Comment: Preprint version of a paper accepted for publication in SIAM Journal
on Mathematical Analysis

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## Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2014

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We developed a procedure of reducing the number of vertices and edges of a
given tree, which we call the "tree simplification procedure," without changing
its topological information. Our motivation for developing this procedure was
to reduce computational costs of graph Laplacian eigenvalues of such trees.
When we applied this procedure to a set of trees representing dendritic
structures of retinal ganglion cells of a mouse and computed their graph
Laplacian eigenvalues, we observed two "plateaux" (i.e., two sets of multiple
eigenvalues) in the eigenvalue distribution of each such simplified tree. In
this article, after describing our tree simplification procedure, we analyze
why such eigenvalue plateaux occur in a simplified tree, and explain such
plateaux can occur in a more general graph if it satisfies a certain condition,
identify these two eigenvalues specifically as well as the lower bound to their
multiplicity.

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## Estimates on the number of eigenvalues of two-particle discrete Schr\"odinger operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/2005

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Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ on the
three-dimensional lattice $\Z^3,$ $k$ being the two-particle quasi-momentum,
are considered. An estimate for the number of the eigenvalues lying outside of
the band of $H_{0}(k)$ via the number of eigenvalues of the potential operator
$V$ bigger than the width of the band of $H_{0}(k)$ is obtained. The existence
of non negative eigenvalues below the band of $H_{0}(k)$ is proven for
nontrivial values of the quasi-momentum $k\in \T^3\equiv (-\pi,\pi]^3$,
provided that the operator H(0) has either a zero energy resonance or a zero
eigenvalue. It is shown that the operator $H(k), k\in \T^3,$ has infinitely
many eigenvalues accumulating at the bottom of the band from below if one of
the coordinates $k^{(j)},j=1,2,3,$ of $k\in \T^3$ is $\pi.$; Comment: 12 pages

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## Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/04/2015

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We consider the solution of large-scale nonlinear algebraic Hermitian
eigenproblems of the form $T(\lambda)v=0$ that admit a variational
characterization of eigenvalues. These problems arise in a variety of
applications and are generalizations of linear Hermitian eigenproblems
$Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally Minimal
Residual (PLMR) method for efficiently computing interior eigenvalues of
problems of this type. We discuss the development of search subspaces,
preconditioning, and eigenpair extraction procedure based on the refined
Rayleigh-Ritz projection. Extension to the block methods is presented, and a
moving-window style soft deflation is described. Numerical experiments
demonstrate that PLMR methods provide a rapid and robust convergence towards
interior eigenvalues. The approach is also shown to be efficient and reliable
for computing a large number of extreme eigenvalues, dramatically outperforming
standard preconditioned conjugate gradient methods.

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## Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We investigate the asymptotic behavior of the eigenvalues of spiked
perturbations of Wigner matrices when the dimension goes to infinity. The
entries of the Hermitian Wigner matrix have a distribution which is symmetric
and satisfies a Poincar\'e inequality. The perturbation matrix is a
deterministic Hermitian matrix whose spectral measure converges to some
probability measure with compact support. We assume that this perturbation
matrix has a fixed number of fixed eigenvalues (spikes) outside the support of
its limiting spectral measure whereas the distance between the other
eigenvalues and this support uniformly goes to zero as the dimension goes to
infinity. We establish that only a particular subset of the spikes will
generate some eigenvalues of the deformed model which will converge to some
limiting points outside the support of the limiting spectral measure. This
phenomenon can be fully described in terms of free probability involving the
subordination function related to the additive free convolution of the limiting
spectral measure of the perturbation matrix by a semi-circular distribution.
Note that up to now only finite rank perturbations had been considered (even in
the deformed GUE case).

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## Semicontinuity of Eigenvalues under Flat Convergence in Euclidean Space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/09/2012

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Recall that Federer-Fleming defined the notion of flat convergence of
submanifolds of Euclidean space to solve the Plateau problem. Here we prove the
upper semicontinuity of Neumann eigenvalues of the submanifolds when they
converge in the flat sense without losing volume. With an additional condition
on the boundaries of the submanifolds we prove the Dirichlet eigenvalues are
semicontinuous as well. We show this additional boundary condition is necessary
as well as the condition that the volumes converge to the volume of the limit
submanifold. As an application of our theorems we see that the Dirichlet and
Neumann eigenvalues of a sequence of surfaces with a common smooth boundary
curve approaching the solution to the Plateau problem are upper semicontinuous.
This work is built upon Fukaya's study of the metric measure convergence of
Riemannian manifolds. One may recall that Cheeger-Colding proved continuity of
the eigenvalues when manifolds with uniform lower Ricci curvature bounds
converge in the metric measure sense. While they obtain continuity, here, we
produce an example demonstrating that continuity is impossible to obtain with
our weaker hypothesis.; Comment: 13 pages, 3 figures

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## Explicit eigenvalues of certain scaled trigonometric matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In a very recent paper "\emph{On eigenvalues and equivalent transformation of
trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78
(2012)), the authors motivated and discussed a trigonometric matrix that arises
in the design of finite impulse response (FIR) digital filters. The eigenvalues
of this matrix shed light on the FIR filter design, so obtaining them in closed
form was investigated. Zhang \emph{et al.}\ proved that their matrix had rank-4
and they conjectured closed form expressions for its eigenvalues, leaving a
rigorous proof as an open problem. This paper studies trigonometric matrices
significantly more general than theirs, deduces their rank, and derives
closed-forms for their eigenvalues. As a corollary, it yields a short proof of
the conjectures in the aforementioned paper.; Comment: 7 pages; fixed Lemma 2, tightened inequalities

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## Guaranteed Lower and upper bounds for eigenvalues of second order elliptic operators in any dimension

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/06/2014

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In this paper, a new method is proposed to produce guaranteed lower bounds
for eigenvalues of general second order elliptic operators in any dimension.
Unlike most methods in the literature, the proposed method only needs to solve
one discrete eigenvalue problem but not involves any base or intermediate
eigenvalue problems, and does not need any a priori information concerning
exact eigenvalues either. Moreover, it just assumes basic regularity of exact
eigenfunctions. This method is defined by a novel generalized Crouzeix-Raviart
element which is proved to yield asymptotic lower bounds for eigenvalues of
general second order elliptic operators, and a simple post-processing method.
As a byproduct, a simple and cheap method is also proposed to obtain guaranteed
upper bounds for eigenvalues, which is based on generalized Crouzeix-Raviart
element approximate eigenfunctions, an averaging interpolation from the the
generalized Crouzeix-Raviart element space to the conforming linear element
space, and an usual Rayleigh-Ritz procedure. The ingredients for the analysis
consist of a crucial projection property of the canonical interpolation
operator of the generalized Crouzeix-Raviart element, explicitly computable
constants for two interpolation operators. Numerics are provided to demonstrate
the theoretical results.

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## Eigenvalues of harmonic almost submersions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/09/2008

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Maps between Riemannian manifolds which are submersions on a dense subset,
are studied by means of the eigenvalues of the pull-back of the target metrics,
the first fundamental form. Expressions for the derivatives of these
eigenvalues yield characterizations of harmonicity, totally geodesic maps and
biconformal changes of metric preserving harmonicity. A Schwarz lemma for
pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues
and, in dimension five, a Bochner technique method, involving the Laplacian of
the difference of the eigenvalues, gives conditions forcing pseudo harmonic
morphisms to be harmonic morphisms.; Comment: 29 pages

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## Outlier eigenvalues for deformed i.i.d. random matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We consider a square random matrix of size N of the form A + Y where A is
deterministic and Y has iid entries with variance 1/N. Under mild assumptions,
as N grows, the empirical distribution of the eigenvalues of A+Y converges
weakly to a limit probability measure \beta on the complex plane. This work is
devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the
complement of the support of \beta. Even in the simplest cases, a variety of
interesting phenomena can occur. As in earlier works, we give a sufficient
condition to guarantee that outliers are stable and provide examples where
their fluctuations vary with the particular distribution of the entries of Y or
the Jordan decomposition of A. We also exhibit concrete examples where the
outlier eigenvalues converge in distribution to the zeros of a Gaussian
analytic function.; Comment: Introduction developed and minor corrections

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## The WKB approximation of semiclassical eigenvalues of the Zakharov-Shabat problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/10/2013

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We numerically compute eigenvalues of the non-self-adjoint Zakharov--Shabat
problem in the semiclassical regime. In particular, we compute the eigenvalues
for a Gaussian potential and compare the results to the corresponding (formal)
WKB approximations used in the approach to the semiclassical or zero-dispersion
limit of the focusing nonlinear Schroedinger equation via semiclassical soliton
ensembles. This numerical experiment, taken together with recent numerical
experiments [17,18], speaks directly to the viability of this approach; in
particular, our experiment suggests a value for the rate of convergence of the
WKB eigenvalues to the true eigenvalues in the semiclassical limit. This
information provides some hint as to how these approximations might be
rigorously incorporated into the asymptotic analysis of the singular limit for
the associated nonlinear partial differential equation.; Comment: 21 pages, 4 figures

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## Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We compute exact asymptotic of the statistical density of random matrices
belonging to the Generalized Gaussian orthogonal, unitary and symplectic
ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$.
In particular, we show that the probability that all the eigenvalues of an
$(n\times n)$ random matrix are positive (negative) decreases for large $n$ as
$\sim exp[-\beta\theta(\alpha)n^2]$ where the Dyson index $\beta$ characterizes
the ensemble, $\alpha$ is some extra parameter and the exponent
$\theta(\alpha)$ is a function of $\alpha$ which will be given explicitly. For
$\alpha=0$, $\theta(0)= (\log 3)/4 = 0.274653...$ is universal. We compute the
probability that the eigenvalues lie in the interval $[\sigma,+\infty[$ with
$(\sigma>0,\; {\rm if}\;\alpha>0)$ and $(\sigma\in\mathbb R,\; {\rm if
}\;\alpha=0)$. This generalizing the celebrated Wigner semicircle law to these
restricted ensembles. It is found that the density of eigenvalues generically
exhibits an inverse square-root singularity at the location of the barriers.
These results generalized the case of Gaussian random matrices ensemble studied
in \cite{D}, \cite{S}.; Comment: arXiv admin note: text overlap with arXiv:0801.1730 by other authors

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## Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2015

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In this paper, inequalities among eigenvalues of different self-adjoint
discrete Sturm-Liouville problems are established. For a fixed discrete
Sturm-Liouville equation, inequalities among eigenvalues for different boundary
conditions are given. For a fixed boundary condition, inequalities among
eigenvalues for different equations are given. These results are obtained by
applying continuity and discontinuity of the n-th eigenvalue function,
monotonicity in some direction of the n-th eigenvalue function, which were
given in our previous papers, and natural loops in the space of boundary
conditions. Some results generalize the relevant existing results about
inequalities among eigenvalues of different Sturm-Liouville problems.; Comment: 32 pages, 5 figures

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## On the eigenvalues of certain Cayley graphs and arrangement graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In this paper, we show that the eigenvalues of certain classes of Cayley
graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all
the k-permutations of an n-element set as vertices where two k-permutations are
adjacent if they differ in exactly r positions. We establish a relation between
the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley
graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1)
follows.; Comment: 12 pages, final version

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## Descriptor approach for eliminating spurious eigenvalues in hydrodynamic equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We describe a general framework for avoiding spurious eigenvalues --
unphysical unstable eigenvalues that often occur in hydrodynamic stability
problems. In two example problems, we show that when system stability is
analyzed numerically using {\em descriptor} notation, spurious eigenvalues are
eliminated. Descriptor notation is a generalized eigenvalue formulation for
differential-algebraic equations that explicitly retains algebraic constraints.
We propose that spurious eigenvalues are likely to occur when algebraic
constraints are used to analytically reduce the number of independent variables
in a differential-algebraic system of equations before the system is
approximated numerically. In contrast, the simple and easily generalizable
descriptor framework simultaneously solves the differential equations and
algebraic constraints and is well-suited to stability analysis in these
systems.; Comment: 13 pages, 1 figure, revised for submission to SIAM Sci. Comp., moved
background information to appendices

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## Inequalities for eigenvalues of the weighted Hodge Laplacian

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/12/2013

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In this paper, we obtain "universal" inequalities for eigenvalues of the
weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These
inequalities generalize the Yang-type and Levitin-Parnovski inequalities for
eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng
and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the
weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.; Comment: 19 pages,any comments are welcome

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## On the distribution of eigenvalues of Maass forms on certain moonshine groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/01/2013

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In this paper we study, both analytically and numerically, questions
involving the distribution of eigenvalues of Maass forms on the moonshine
groups $\Gamma_0(N)^+$, where N>1$ is a square-free integer. After we prove
that $\Gamma_0(N)^+$ has one cusp, we compute the constant term of the
associated non-holomorphic Eisenstein series. We then derive an "average"
Weyl's law for the distribution of eigenvalues of Maass forms, from which we
prove the "classical" Weyl's law as a special case. The groups corresponding to
N=5 and N=6 have the same signature; however, our analysis shows that,
asymptotically, there are infinitely more cusp forms for $\Gamma_0(5)^+$ than
for $\Gamma_0(6)^+$. We view this result as being consistent with the
Phillips-Sarnak philosophy since we have shown, unconditionally, the existence
of two groups which have different Weyl's laws. In addition, we employ Hejhal's
algorithm, together with recently developed refinements from [31], and
numerically determine the first 3557 of $\Gamma_0(5)^+$ and the first 12474
eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify
some conjectured distributional properties of the eigenvalues.; Comment: A version with higher resolution figures can be downloaded from
http://www.maths.bris.ac.uk/~mahlt/research/JST2012a.pdf

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## On the eigenvalues of some nonhermitian oscillators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/01/2013

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We consider a class of one-dimensional nonhermitian oscillators and discuss
the relationship between the real eigenvalues of PT-symmetric oscillators and
the resonances obtained by different authors. We also show the relationship
between the strong-coupling expansions for the eigenvalues of those
oscillators. Comparison of the results of the complex rotation and the
Riccati-Pad\'{e} methods reveals that the optimal rotation angle converts the
oscillator into either a PT-symmetric or an Hermitian one. In addition to the
real positive eigenvalues the PT-symmetric oscillators exhibit real positive
resonances under different boundary conditions. They can be calculated by means
of the straightforward diagonalization method. The Riccati-Pad\'e method yields
not only the resonances of the nonhermitian oscillators but also the
eigenvalues of the PT-symmetric ones.

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## Analysis of structural correlations in a model binary 3D liquid through the eigenvalues and eigenvectors of the atomic stress tensors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/10/2015

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It is possible to associate with every atom or molecule in a liquid its own
atomic stress tensor. These atomic stress tensors can be used to describe
liquids' structures and to investigate the connection between structural and
dynamic properties. In particular, atomic stresses allow to address atomic
scale correlations relevant to the Green-Kubo expression for viscosity.
Previously correlations between the atomic stresses of different atoms were
studied using the Cartesian representation of the stress tensors or the
representation based on spherical harmonics. In this paper we address
structural correlations in a model 3D binary liquid using the eigenvalues and
eigenvectors of the atomic stress tensors. Thus correlations relevant to the
Green-Kubo expression for viscosity are interpreted in a simple geometric way.
On decrease of temperature the changes in the relevant stress correlation
function between different atoms are significantly more pronounced than the
changes in the pair density function. We demonstrate that this behaviour
originates from the orientational correlations between the eigenvectors of the
atomic stress tensors. We also found correlations between the eigenvalues of
the same atomic stress tensor. For the studied system...

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## Exponential Decay of Eigenfunctions and Accumulation of Eigenvalues on Manifolds with Axial Analytic Asymptotically Cylindrical Ends

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/07/2010

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#Mathematics - Spectral Theory#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Differential Geometry#58J50, 58J05, 58J32

In this paper we continue our study of the Laplacian on manifolds with axial
analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using
the complex scaling method and the Phragm\'{e}n-Lindel\"{o}f principle we prove
exponential decay of the eigenfunctions corresponding to the non-threshold
eigenvalues of the Laplacian on functions. In the case of a manifold with
(non-compact) boundary it is either the Dirichlet Laplacian or the Neumann
Laplacian. We show that the rate of exponential decay of an eigenfunction is
prescribed by the distance from the corresponding eigenvalue to the next
threshold. Under our assumptions on the behaviour of the metric at infinity
accumulation of isolated and embedded eigenvalues occur. The results on decay
of eigenfunctions combined with the compactness argument due to Perry imply
that the eigenvalues can accumulate only at thresholds and only from below. The
eigenvalues are of finite multiplicity.; Comment: 33 pages, 4 figures

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