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## Transmission eigenvalues for operators with constant coefficients

Hitrik, Michael; Krupchyk, Katsiaryna; Ola, Petri; Päivärinta, Lassi
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

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

McKee, James; Smyth, Chris
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Aktosun, Tuncay; Gintides, Drossos; Papanicolaou, Vassilis G.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Demengel, Francoise
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Strohmaier, Alexander; Uski, Ville
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Li, Xueliang; Wang, Jianfeng; Huang, Qiongxiang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Tanaka, Tohru; Ota, Yukihiro; Kanazawa, Mitsunori; Kimura, Gen; Nakazato, Hiromichi; Nori, Franco
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

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

Gittens, Alex; Tropp, Joel A.
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...

## Eigenvalues of the Laplacian on Riemannian manifolds

Cheng, Qing-Ming; Qi, Xuerong
Tipo: Artigo de Revista Científica
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

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

Astrauskas, Arvydas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

## Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

Dean, David S.; Majumdar, Satya N.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

## Strong Linear Correlation Between Eigenvalues and Diagonal Matrix Elements

Shen, J. J.; Arima, A.; Zhao, Y. M.; Yoshinaga, N.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

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

Cvetkovic, Dragos; Grout, Jason
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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

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

Benaych-Georges, Florent; Guionnet, Alice; Maïda, Mylène
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

## Perturbations of embedded eigenvalues for the planar bilaplacian

Derks, Gianne; Sasane, Sara Maad; Sandstede, Bjorn
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

## On the extreme eigenvalues of regular graphs

Cioaba, Sebastian M.
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

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

Baik, Jinho; Silverstein, Jack W.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.6%
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

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

Ilias, Said; Makhoul, Ola
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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)

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

Albeverio, Sergio; Lakaev, Saidakhmat N.; Rasulov, Tulkin H.
Tipo: Artigo de Revista Científica
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