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## Transmission eigenvalues for operators with constant coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2010

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26.6%

In this paper we study the interior transmission problem and transmission
eigenvalues for multiplicative perturbations of linear partial differential
operator of order $\ge 2$ with constant real coefficients. Under suitable
growth conditions on the symbol of the operator and the perturbation, we show
the discreteness of the set of transmission eigenvalues and derive sufficient
conditions on the existence of transmission eigenvalues. We apply these
techniques to the case of the biharmonic operator and the Dirac system. In the
hypoelliptic case we present a connection to scattering theory.

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## Integer symmetric matrices having all their eigenvalues in the interval [-2,2]

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/05/2007

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We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).; Comment: 33 pages, 18 figures

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## The uniqueness in the inverse problem for transmission eigenvalues for the spherically-symmetric variable-speed wave equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/06/2011

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The recovery of a spherically-symmetric wave speed $v$ is considered in a
bounded spherical region of radius $b$ from the set of the corresponding
transmission eigenvalues for which the corresponding eigenfunctions are also
spherically symmetric. If the integral of $1/v$ on the interval $[0,b]$ is less
than $b,$ assuming that there exists at least one $v$ corresponding to the
data, it is shown that $v$ is uniquely determined by the data consisting of
such transmission eigenvalues and their "multiplicities," where the
"multiplicity" is defined as the multiplicity of the transmission eigenvalue as
a zero of a key quantity. When that integral is equal to $b,$ the unique
recovery is obtained when the data contains one additional piece of
information. Some similar results are presented for the unique determination of
the potential from the transmission eigenvalues with "multiplicities" for a
related Schr\"odinger equation.; Comment: 30 pages, no figures

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## Generalized eigenvalues for fully tnonlinear singular or degenerate operators in the radial case

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/04/2009

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In this paper we extend some existence's results concerning the generalized
eigenvalues for fully nonlinear operators singular or degenerate. We consider
the radial case and we prove the existence of an infinite number of
eigenvalues, simple and isolated. This completes the results obtained by the
author with Isabeau Birindelli for the first eigenvalues in the radial case,
and the results obtained for the Pucci's operator by Busca Esteban and Quaas
and for the $p$-Laplace operator by Del Pino and Manasevich.; Comment: 34 pages, no figures

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## An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Spectral Theory#Mathematical Physics#Mathematics - Numerical Analysis#11F72, 11M36, 35P15, 58J52

We present a rigorous scheme that makes it possible to compute eigenvalues of
the Laplace operator on hyperbolic surfaces within a given precision. The
method is based on an adaptation of the method of particular solutions to the
case of locally symmetric spaces and on explicit estimates for the
approximation of eigenfunctions on hyperbolic surfaces by certain basis
functions. It can be applied to check whether or not there is an eigenvalue in
an \epsilon-neighborhood of a given number \lambda>0. This makes it possible to
find all the eigenvalues in a specified interval, up to a given precision with
rigorous error estimates. The method converges exponentially fast with the
number of basis functions used. Combining the knowledge of the eigenvalues with
the Selberg trace formula we are able to compute values and derivatives of the
spectral zeta function again with error bounds. As an example we calculate the
spectral determinant and the Casimir energy of the Bolza surface and other
surfaces.; Comment: 48 pages, 8 figures, LaTeX, some more typos corrected, more Figures
added, some explanations are more detailed now

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## On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/05/2014

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Harary and Schwenk posed the problem forty years ago: Which graphs have
distinct adjacency eigenvalues? In this paper, we obtain a necessary and
sufficient condition for an Hermitian matrix with simple spectral radius and
distinct eigenvalues. As its application, we give an algebraic characterization
to the Harary-Schwenk's problem. As an extension of their problem, we also
obtain a necessary and sufficient condition for a positive semidefinite matrix
with simple least eigenvalue and distinct eigenvalues, which can provide an
algebraic characterization to their problem with respect to the (normalized)
Laplacian matrix.; Comment: 11 pages

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## Determining eigenvalues of a density matrix with minimal information in a single experimental setting

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.6%

Eigenvalues of a density matrix characterize well the quantum state's
properties, such as coherence and entanglement. We propose a simple method to
determine all the eigenvalues of an unknown density matrix of a
finite-dimensional system in a single experimental setting. Without fully
reconstructing a quantum state, eigenvalues are determined with the minimal
number of parameters obtained by a measurement of a single observable.
Moreover, its implementation is illustrated in linear optical and
superconducting systems.; Comment: 5 pages, 2 figures

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## Tail bounds for all eigenvalues of a sum of random matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.6%

This work introduces the minimax Laplace transform method, a modification of
the cumulant-based matrix Laplace transform method developed in "User-friendly
tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both
upper and lower bounds on each eigenvalue of a sum of random self-adjoint
matrices. This machinery is used to derive eigenvalue analogues of the
classical Chernoff, Bennett, and Bernstein bounds.
Two examples demonstrate the efficacy of the minimax Laplace transform. The
first concerns the effects of column sparsification on the spectrum of a matrix
with orthonormal rows. Here, the behavior of the singular values can be
described in terms of coherence-like quantities. The second example addresses
the question of relative accuracy in the estimation of eigenvalues of the
covariance matrix of a random process. Standard results on the convergence of
sample covariance matrices provide bounds on the number of samples needed to
obtain relative accuracy in the spectral norm, but these results only guarantee
relative accuracy in the estimate of the maximum eigenvalue. The minimax
Laplace transform argument establishes that if the lowest eigenvalues decay
sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples...

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## Eigenvalues of the Laplacian on Riemannian manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/04/2011

Relevância na Pesquisa

26.6%

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete
Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue
problem of the Laplacian. By making use of a fact that eigenfunctions form an
orthonormal basis of $L^2(\Omega)$ in place of the Rayleigh-Ritz formula, we
obtain inequalities for eigenvalues of the Laplacian. In particular, for lower
order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.; Comment: 17 pages

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## From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/01/2015

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The aim of this paper is to study asymptotic geometric properties almost
surely or/and in probability of extreme order statistics of an i.i.d. random
field (potential) indexed by sites of multidimensional lattice cube, the volume
of which unboundedly increases. We discuss the following topics: (I) high level
exceedances, in particular, clustering of exceedances; (II) decay rate of
spacings in comparison with increasing rate of extreme order statistics; (III)
minimum of spacings of successive order statistics; (IV) asymptotic behavior of
values neighboring to extremes and so on. The conditions of the results are
formulated in terms of regular variation (RV) of the cumulative hazard function
and its inverse. A relationship between RV classes of the present paper as well
as their links to the well-known RV classes (including domains of attraction of
max-stable distributions) are discussed.
The asymptotic behavior of functionals (I)--(IV) determines the asymptotic
structure of the top eigenvalues and the corresponding eigenfunctions of the
large-volume discrete Schr\" odinger operators with an i.i.d. potential
(Anderson Hamiltonian). Thus, another aim of the present paper is to review and
comment a recent progress on extreme value theory for eigenvalues of random
Schr\"odinger operators as well as to provide a clear and rigorous
understanding of the relationship between the top eigenvalues and extreme
values of i.i.d. potentials.; Comment: 55 pages

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## Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/01/2008

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26.6%

We compute exact asymptotic results for the probability of the occurrence of
large deviations of the largest (smallest) eigenvalue of random matrices
belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In
particular, we show that the probability that all the eigenvalues of an (NxN)
random matrix are positive (negative) decreases for large N as ~\exp[-\beta
\theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the
exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the
probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which
allows us to calculate the joint probability distribution of the minimum and
the maximum eigenvalue. As a byproduct, we also obtain exactly the average
density of states in Gaussian ensembles whose eigenvalues are restricted to lie
in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner
semi-circle law to these restricted ensembles. It is found that the density of
states generically exhibits an inverse square-root singularity at the location
of the barriers. These results are confirmed by numerical simulations.; Comment: 17 pages Revtex, 5 .eps figures included

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## Strong Linear Correlation Between Eigenvalues and Diagonal Matrix Elements

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/01/2008

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26.6%

We investigate eigenvalues of many-body systems interacting by two-body
forces as well as those of random matrices. We find a strong linear correlation
between eigenvalues and diagonal matrix elements if both of them are sorted
from the smaller values to larger ones. By using this linear correlation we are
able to predict reasonably all eigenvalues of given shell model Hamiltonian
without complicated iterations.; Comment: 4 pages

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## Graphs with extremal energy should have a small number of distinct eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/10/2007

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26.6%

The sum of the absolute values of the eigenvalues of a graph is called the
energy of the graph. We study the problem of finding graphs with extremal
energy within specified classes of graphs. We develop tools for treating such
problems and obtain some partial results. Using calculus, we show that an
extremal graph ``should'' have a small number of distinct eigenvalues. However,
we also present data that shows in many cases that extremal graphs can have a
large number of distinct eigenvalues.; Comment: 17 pages; contains a SAGE program and minor grammatical corrections
that are not contained in the published version

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## Large deviations of the extreme eigenvalues of random deformations of matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.6%

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral
measure converging to a compactly supported probability measure. We perturb
this matrix by adding a random finite rank matrix, with delocalized
eigenvectors. We show that the joint law of the extreme eigenvalues of the
perturbed model satisfies a large deviation principle in the scale $n$, with a
good rate function given by a variational formula. We tackle both cases when
the extreme eigenvalues of $X_n$ converge to the edges of the support of the
limiting measure and when we allow some eigenvalues of $X_n$, that we call
outliers, to converge out of the bulk. We can also generalise our results to
the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud
X,$ for $V$ growing fast enough at infinity and any perturbation of finite
rank.; Comment: 44 pages

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## Perturbations of embedded eigenvalues for the planar bilaplacian

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/09/2010

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26.6%

Operators on unbounded domains may acquire eigenvalues that are embedded in
the essential spectrum. Determining the fate of these embedded eigenvalues
under small perturbations of the underlying operator is a challenging task, and
the persistence properties of such eigenvalues is linked intimately to the
multiplicity of the essential spectrum. In this paper, we consider the planar
bilaplacian with potential and show that the set of potentials for which an
embedded eigenvalue persists is locally an infinite-dimensional manifold with
infinite codimension in an appropriate space of potentials.

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## On the extreme eigenvalues of regular graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.6%

In this paper, we present an elementary proof of a theorem of Serre
concerning the greatest eigenvalues of $k$-regular graphs. We also prove an
analogue of Serre's theorem regarding the least eigenvalues of $k$-regular
graphs: given $\epsilon>0$, there exist a positive constant $c=c(\epsilon,k)$
and a nonnegative integer $g=g(\epsilon,k)$ such that for any $k$-regular graph
$X$ with no odd cycles of length less than $g$, the number of eigenvalues $\mu$
of $X$ such that $\mu \leq -(2-\epsilon)\sqrt{k-1}$ is at least $c|X|$. This
implies a result of Winnie Li.; Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some
comments on the constant c in Section 2

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## Eigenvalues of Large Sample Covariance Matrices of Spiked Population Models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/08/2004

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We consider a spiked population model, proposed by Johnstone, whose
population eigenvalues are all unit except for a few fixed eigenvalues. The
question is to determine how the sample eigenvalues depend on the non-unit
population ones when both sample size and population size become large. This
paper completely determines the almost sure limits for a general class of
samples.; Comment: 24 pages, 6 figures

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## Universal inequalities for the eigenvalues of a power of the Laplace operator

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/01/2010

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26.6%

In this paper, we obtain a new abstract formula relating eigenvalues of a
self-adjoint operator to two families of symmetric and skew-symmetric operators
and their commutators. This formula generalizes earlier ones obtained by
Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how
one can use this abstract formulation both for giving dierent and simpler
proofs for all the known results obtained for the eigenvalues of a power of the
Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for
the bilaplacian and more generally for the polyharmonic problem on a bounded
Euclidean domain) and to obtain new ones. In a last paragraph, we derive new
bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg
group.; Comment: A para\^itre \`a Manuscripta Mathematica (29 pages)

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## The number of eigenvalues for an Hamiltonian in Fock space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/2005

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A model operator $H$ corresponding to the energy operator of a system with
non-conserved number $n\leq 3$ of particles is considered. The precise location
and structure of the essential spectrum of $H$ is described. The existence of
infinitely many eigenvalues below the bottom of the essential spectrum of $H$
is proved if the generalized Friedrichs model has a virtual level at the bottom
of the essential spectrum and for the number $N(z)$ of eigenvalues below $z<0$
an asymptotics established. The finiteness of eigenvalues of $H$ below the
bottom of the essential spectrum is proved if the generalized Friedrichs model
has a zero eigenvalue at the bottom of its essential spectrum.; Comment: 23 pages

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## Poisson Statistics for Eigenvalues of Continuum Random Schr\"odinger Operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We show absence of energy levels repulsion for the eigenvalues of random
Schr\"odinger operators in the continuum. We prove that, in the localization
region at the bottom of the spectrum, the properly rescaled eigenvalues of a
continuum Anderson Hamiltonian are distributed as a Poisson point process with
intensity measure given by the density of states. We also obtain simplicity of
the eigenvalues. We derive a Minami estimate for continuum Anderson
Hamiltonians. We also give a simple and transparent proof of Minami's estimate
for the (discrete) Anderson model.; Comment: updated references, misprints corrected

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