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Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues
Michael Levitin,Leonid Parnovski
Mathematics , 2001, DOI: 10.1006/jfan.2001.3913
Abstract: Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian (Payne-Polya-Weinberger, Hile-Protter, etc.), as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.
Spectral pollution and second order relative spectra for self-adjoint operators
Michael Levitin,Eugene Shargorodsky
Mathematics , 2002,
Abstract: We consider the phenomenon of spectral pollution arising in calculation of spectra of self-adjoint operators by projection methods. We suggest a strategy of dealing with spectral pollution by using the so-called second order relative spectra. The effectiveness of the method is illustrated by a detailed analysis of two model examples.
On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators
Lyonell Boulton,Michael Levitin
Mathematics , 2007, DOI: 10.1088/1751-8113/40/31/010
Abstract: This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called quadratic projection method, in order to achieve convergence free from spectral pollution. We describe the theoretical foundations of the method in detail, and illustrate its effectiveness by several examples.
A simple method of calculating eigenvalues and resonances in domains with infinite regular ends
Michael Levitin,Marco Marletta
Mathematics , 2006,
Abstract: A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev [{\em Q. J. Mech. Appl. Math.} {\bfseries 53} (2000), 429--447].
Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential
Michael Levitin,Marcello Seri
Mathematics , 2015,
Abstract: For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm--Liouville operator $A = \mathrm{sign}(x)(-\Delta + V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.
Victor Borisovich Lidskii (1924-2008)
Michael Levitin,Dmitri Vassiliev
Mathematics , 2010,
Abstract: This is the editors' preface to the volume "Operator theory and its applications, in memory of V.B.Lidskii (1924-2008)". The volume is due to be published by the American Mathematical Society in the series American Mathematical Society Translations, series 2, volume 231.
Range of the first three eigenvalues of the planar Dirichlet Laplacian
Michael Levitin,Rustem Yagudin
Mathematics , 2002,
Abstract: We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio $\lambda_3/\lambda_1$.
On the principal eigenvalue of a Robin problem with a large parameter
Michael Levitin,Leonid Parnovski
Mathematics , 2004,
Abstract: We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.
Spectra of a class of non-self-adjoint matrices
E. Brian Davies,Michael Levitin
Mathematics , 2013, DOI: 10.1016/j.laa.2014.01.025
Abstract: We consider a new class of non-self-adjoint matrices that arise from an indefinite self-adjoint linear pencil of matrices, and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity. We prove that the spectrum is qualitatively different when a certain parameter $c$ equals $0$, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of $c$.
Isospectral domains with mixed boundary conditions
Michael Levitin,Leonid Parnovski,Iosif Polterovich
Mathematics , 2005, DOI: 10.1088/0305-4470/39/9/006
Abstract: We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M. Kac.
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