Books and Editorial

Michael Levitin, Dan Mangoubi, and Iosif Polterovich, Topics in Spectral Geometry
see the separate page
Geometric and Computational Spectral Theory, edited by Alexandre Girouard, Dmitry Jakobson, Michael Levitin, Nilima Nigam, Iosif Polterovich, Frédéric Rochon, AMS Contemporary Mathematics Series, volume 700, 2017; 296 pp.
doidoi: 10.1090/conm/700
▼ Abstract
▲ Abstract A co-publication of the AMS and Centre de Recherches Mathématiques. The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 Séminaire de Mathématiques Supérieures on Geometric and Computational Spectral Theory, held from June 15–26, 2015, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.
Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008), edited by M Levitin and D Vassiliev, American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences, volume 231, 2010; 180 pp.
doidoi: 10.1090/trans2/231
▼ Abstract
▲ Abstract This book is a collection of articles devoted to the theory of linear operators in Hilbert spaces and its applications. The subjects covered range from the abstract theory of Toeplitz operators to the analysis of very specific differential operators arising in quantum mechanics, electromagnetism, and the theory of elasticity; the stability of numerical methods is also discussed. Many of the articles deal with spectral problems for not necessarily selfadjoint operators. Some of the articles are surveys outlining the current state of the subject and presenting open problems.
L Boulton and M Levitin, Trends and Tricks in Spectral Theory, Ediciones IVIC, Caracas, 2007; 99 pp. ISBN 978-980-261-086-0.

 

Papers and Preprints

Where available, links to the final published texts are provided via doi; archived preprints (also linked) may differ from final versions.

 

Preprints and papers in press

N Filonov, M Levitin, I Polterovich, and D A Sher, Uniform enclosures for the phase and zeros of Bessel functions and their derivatives, 2024.
PdfPDF (arXiv:math) and accompanying script in a separate page
▼ Abstract
▲ Abstract We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.
N Filonov, M Levitin, I Polterovich, and D A Sher, Inequalities à la Pólya for the Aharonov–Bohm eigenvalues of the disk, 2023. To appear in Journal of Spectral Theory.
PdfPDF (arXiv:math) and accompanying script in a separate page
▼ Abstract
▲ Abstract We prove an analogue of Pólya's conjecture for the eigenvalues of the magnetic Schrödinger operator with Aharonov--Bohm potential on the disk, for Dirichlet and magnetic Neumann boundary conditions. This answers a question posed by R. L. Frank and A. M. Hansson in 2008.

 

Published papers

N Filonov, M Levitin, I Polterovich, and D A Sher, Pólya’s conjecture for Euclidean balls, Inventiones Mathematicae 234 (2023), 129-169.
PdfPDF (arXiv:math) and accompanying script in a separate page
doi: doi10.1007/s00222-023-01198-1
▼ Abstract
▲ Abstract The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.
M Capoferri, L Friedlander, M Levitin, and D Vassiliev, Two-term spectral asymptotics in linear elasticity, Journal of Geometric Analysis 33 (2023), article 242.
PdfPDF (arXiv:math)
doi: doi10.1007/s12220-023-01269-y
▼ Abstract
▲ Abstract We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published in [J. Geom. Anal. 31 (2021), 10164–10193].
M Capoferri, M Levitin, and D Vassiliev, Geometric wave propagator on Riemannian manifolds, Communications in Analysis and Geometry 30:8 (2022), 1713-1777.
PdfPDF (arXiv:math)
doi: doi10.4310/CAG.2022.v30.n8.a2
▼ Abstract
▲ Abstract We study the propagator of the wave equation on a closed Riemannian manifold M. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator - a scalar function on the cotangent bundle - and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise topological obstructions and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.
M Levitin, L Parnovski, I Polterovich, and D A Sher, Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons, Proceedings of the LMS 125:3 (2022), 359-487.
PdfPDF (arXiv:math)
doi: doi10.1112/plms.12461
▼ Abstract
▲ Abstract We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.
A Girouard, M Karpukhin, M Levitin, and I Polterovich, The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript, Journal of Spectral Theory 12:1 (2022), 195-225.
PdfPDF (arXiv:math)
doi: doi10.4171/JST/399
▼ Abstract
▲ Abstract How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.
M Levitin, L Parnovski, I Polterovich, and D A Sher, Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues, Journal d'Analyse Mathématique 146 (2022), 65-125.
PdfPDF (arXiv:math)
doi: doi10.1007/s11854-021-0188-x
▼ Abstract
▲ Abstract In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.
M Levitin, P Monk, and V Selgas, Impedance eigenvalues in linear elasticity, SIAM Journal on Applied Mathematics 81:6 (2021), 2433-2456.
PdfPDF (arXiv:math)
doi: doi10.1137/21M1412955
▼ Abstract
▲ Abstract This paper is devoted to studying impedance eigenvalues (that is, eigenvalues of a particular Dirichlet-to-Neumann map) for the time harmonic linear elastic wave problem, and their potential use as target-signatures for fluid-solid interaction problems. We first consider several possible families of eigenvalues of the elasticity problem, focusing on certain impedance eigenvalues that are an analogue of Steklov eigenvalues. We show that one of these families arises naturally in inverse scattering. We also analyse their approximation from far field measurements of the scattered pressure field in the fluid, and illustrate several alternative methods of approximation in the case of an isotropic elastic disk.
M Levitin and A Strohmaier, Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cusps, International Mathematics Research Notices 2021:6 (2021), 4003-4050.
PdfPDF (arXiv:math) and accompanying videos on a separate page or on YouTube
doi: doi10.1093/imrn/rnz157
▼ Abstract
▲ Abstract In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.
S Krymski, M Levitin, L Parnovski, I Polterovich, and D A Sher, Inverse Steklov spectral problem for curvilinear polygons, International Mathematics Research Notices 2021:1 (2021), 1-37.
PdfPDF (arXiv:math)
doi: doi10.1093/imrn/rnaa200
▼ Abstract
▲ Abstract This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than π, we prove that the asymptotics of Steklov eigenvalues obtained in Levitin, Parnovski, Polterovich, Sher (2019) determines, in a constructive manner, the number of vertices, and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra.
M Levitin and H Öztürk, A two-parameter eigenvalue problem for a class of block-operator matrices, in The Diversity and Beauty of Applied Operator Theory, Operator Theory: Advances and Applications 268, 2018, Birkhäuser, Basel.
PdfPDF (arXiv:math)
doi: doi10.1007/978-3-319-75996-8_19
▼ Abstract
▲ Abstract We consider a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, we state localisation theorems for the pair-eigenvalues and discuss relations to a class of non-self-adjoint spectral problems.
Y-L Fang, M Levitin, and D Vassiliev, Spectral analysis of the Dirac operator on a 3-sphere, Operators and Matrices 12 (2018), 501-527.
PdfPDF (arXiv:math)
doi: doi10.7153/oam-2018-12-31
▼ Abstract
▲ Abstract We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrary smooth fashion from the standard one. We derive explicit asymptotic formulae for the two eigenvalues closest to zero. Note that these eigenvalues remain double eigenvalues under perturbations of the metric: they cannot split because of a particular symmetry of the Dirac operator in dimension three (it commutes with the antilinear operator of charge conjugation). Our asymptotic formulae show that in the first approximation our two eigenvalues maintain symmetry about zero and are completely determined by the increment of Riemannian volume. Spectral asymmetry is observed only in the second approximation of the perturbation process. As an example we consider a special family of metrics, the so-called generalized Berger spheres, for which the eigenvalues can be evaluated explicitly.
A R Gover, A Hassannezhad, D Jakobson, and M Levitin, Zero and negative eigenvalues of the conformal Laplacian, Journal of Spectral Theory 6:4 (2016), 793–806.
PdfPDF (arXiv:math)
doi: doi10.4171/JST/142
▼ Abstract
▲ Abstract We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.
M Levitin and M Seri, Accumulation of complex eigenvalues of an indefinite Sturm-Liouville operator with a shifted Coulomb potential, Operators and Matrices 10 (2016), 223–245.
PdfPDF (arXiv:math)
doi: doi10.7153/oam-10-14
▼ Abstract
▲ Abstract For a particular family of long-range potentials V, we prove that the eigenvalues of the indefinite Sturm–Liouville operator A = sign(x)(−Δ+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.
E B Davies and M Levitin, Spectra of a class of non-self-adjoint matrices, Linear Algebra and its Applications 448 (2014), 55–84.
PdfPDF (arXiv:math) and mp4MP4 movie (arXiv:math)
doi: doi10.1016/j.laa.2014.01.025
▼ Abstract
▲ 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.
R J Downes, M Levitin and D Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, Journal of Mathematical Physics 54:11, 111503 (2013).
PdfPDF (arXiv:math)
doidoi: 10.1063/1.4828858
▼ Abstract
▲ Abstract Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.
D M Elton, M Levitin and I Polterovich, Eigenvalues of a one-dimensional Dirac operator pencil, Annales Henri Poincaré 15:12, 2321-2377 (2014).
PdfPDF (arXiv:math)
doidoi: 10.1007/s00023-013-0304-2
▼ Abstract
▲ Abstract We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.
M Levitin and D Vassiliev, Victor Borisovich Lidskii (1924-2008), in Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008), edited by M Levitin and D Vassiliev, American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences, volume 231, 2010, 1-6.
PdfPDF (arXiv:math)
doidoi: 10.1090/trans2/231 (volume)
▼ Abstract
▲ Abstract This is the editors' preface to the volume.
M Levitin, A Sobolev, and D Sobolev, On the near periodicity of eigenvalues of Toeplitz matrices, in Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008), edited by M Levitin and D Vassiliev, American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences, volume 231, 2010, 115-126.
PdfPDF (arXiv:math)
doidoi: 10.1090/trans2/231 (volume)
▼ Abstract
▲ Abstract Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.
L Boulton, M Levitin, and M Marletta, On a class of nonselfadjoint periodic boundary value problems with discrete real spectrum, in Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924-2008), edited by M Levitin and D Vassiliev, American Mathematical Society Translations - Series 2, Advances in the Mathematical Sciences, volume 231, 2010, 59-66.
PdfPDF (arXiv:math)
doidoi: 10.1090/trans2/231 (volume)
▼ Abstract
▲ Abstract In Boulton, Levitin, Marletta (J. Diff. Eqs. 2010) we examined a family of periodic Sturm-Liouville problems with boundary and interior singularities which are highly non-self-adjoint but have only real eigenvalues. We now establish Schatten class properties of the associated resolvent operator.
L Boulton, M Levitin, and M Marletta, On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities, Journal of Differential Equations, 249 (2010), 3081-3098.
PdfPDF (arXiv:math)
doidoi: 10.1016/j.jde.2010.08.010
▼ Abstract
▲ Abstract We prove that all the eigenvalues of a certain highly non-self-adjoint Sturm-Liouville differential operator are real. The results presented are motivated by and extend those recently found by various authors (Benilov et al. (2003), Davies (2007) and Weir (2008)) on the stability of a model describing small oscillations of a thin layer of fluid inside a rotating cylinder.
R Benguria, M Levitin, and L Parnovski, Fourier transform, null variety, and Laplacian's eigenvalues, Journal of Functional Analysis 257 (2009), 2088-2123.
PdfPDF (arXiv:math)
doidoi: 10.1016/j.jfa.2009.06.022
▼ Abstract
▲ Abstract We consider a quantity κ(Ω) - the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximized, among all convex balanced domains Ω⊂Rd of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of κ(Ω). We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.
M Levitin and M Marletta, A simple method of calculating eigenvalues and resonances in domains with infinite regular ends, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 138A (2008), 1043-1065.
PdfPDF (arXiv:math)
doidoi: 10.1017/S0308210506001144
▼ Abstract
▲ Abstract We present a simple new approach to the solution of a wide class of spectral and resonance problems on infinite domains with regular ends, including those found in the study of quantum switches, waveguides, and acoustic scatterers. Our algorithm is part analytical and part numerical and is essentially a combination of four classical approaches (domain decomposition, boundary elements, finite elements and spectral methods) each of which is used in its most natural context.
M Levitin and L Parnovski, On the principal eigenvalue of a Robin problem with a large parameter, Mathematische Nachrichten 281 (2008), 272-281.
PdfPDF (arXiv:math)
doidoi: 10.1002/mana.200510600
▼ Abstract
▲ 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.
L Boulton and M Levitin, On approximation of the eigenvalues of perturbed periodic Schrödinger operators, Journal of Physics A: Mathematical and Theoretical 40 (2007), 9319-9329.
PdfPDF (arXiv:math)
doidoi: 10.1088/1751-8113/40/31/010
▼ Abstract
▲ Abstract This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrödinger 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.
M Levitin, L Parnovski, and I Polterovich, Isospectral domains with mixed boundary conditions, Journal of Physics A: Mathematical and Theoretical 39 (2006), 2073-2082.
PdfPDF (arXiv:math)
doidoi: 10.1088/0305-4470/39/9/006
▼ Abstract
▲ 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.
D Jakobson, M Levitin, N Nadirashvili, and I Polterovich, Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond, Journal of Computational and Applied Mathematics 194 (2006), 141-155.
PdfPDF (arXiv:math)
doidoi: 10.1016/j.cam.2005.06.019
▼ Abstract
▲ Abstract Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some results, conjectures and problems related to this variation on the isospectral theme.
D Jakobson, M Levitin, N Nadirashvili, N Nigam, and I Polterovich, How large can the first eigenvalue be on a surface of genus two?, International Mathematics Research Notices 2005:63 (2005), 3967-3985.
PdfPDF (arXiv:math)
doidoi: 10.1155/IMRN.2005.3967
▼ Abstract
▲ Abstract Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface which is realized as a double branched covering over a sphere. The six ramification points are chosen in such a way that this surface is conformally equivalent to the Bolza surface. We prove that our conjecture follows from a lower bound on the first eigenvalue of a certain mixed Dirichlet-Neumann boundary value problem on a half-disk. The latter can be studied numerically, and we present conclusive evidence supporting the conjecture.
E R Johnson, M Levitin, and L Parnovski, Existence of eigenvalues of a linear operator pencil in a curved waveguide — localized shelf waves on a curved coast, SIAM Journal of Mathematical Analysis 37:5 (2006),1465-1481.
PdfPDF (arXiv:math)
doidoi: 10.1137/040615936
▼ Abstract
▲ Abstract The question of the existence of non-propagating, trapped continental shelf waves (CSWs) along curved coasts reduces mathematically to a spectral problem for a self-adjoint operator pencil in a curved strip. Using methods developed for the waveguide trapped mode problem, we show that such CSWs exist for a wide class of coast curvature and depth profiles.
M Levitin and E Shargorodsky, Spectral pollution and second order relative spectra for self-adjoint operators, IMA Journal of Numerical Analysis 24 (2004), 393-416.
PdfPDF (arXiv:math)
doidoi: 10.1093/imanum/24.3.393
▼ Abstract
▲ 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.
M Levitin and R Yagudin, Range of the first three eigenvalues of the planar Dirichlet Laplacian, LMS Journal of Computational Mathematics 6 (2003), 1-17.
PdfPDF (arXiv:math)
doidoi: 10.1112/S1461157000000346
▼ Abstract
▲ 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 of the third and first eigenvalues.
M Levitin and L Parnovski, Trace identities and universal estimates for eigenvalues of linear pencils, in Elliptic and Parabolic Problems, Proceedings of the IV European Conference (Rolduc/Gaeta 2001), edited by J Bemelmans, etc, World Scientific (2002), 160-164.
PdfPDF (this site)
doidoi: 10.1142/9789812777201_0016
▼ Abstract
▲ Abstract We describe the method of constructing the spectral trace identities and the estimates of eigenvalue gaps for the linear self-adjoint operator pencils A-lambda B.
M Levitin and L Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues, Journal of Functional Analysis 192 (2002), 425-445.
PdfPDF (arXiv:math)
doidoi: 10.1006/jfan.2001.3913
▼ Abstract
▲ 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, 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.
M Levitin, Dirichlet Laplacian, in Encyclopaedia of Mathematics, Kluwer (2000).
PdfPDF (this site)
textText
M S Agranovich, B A Amosov and M Levitin, Spectral problems for the Lamé system with spectral parameter in boundary conditions on smooth or nonsmooth boundary, Russian Journal of Mathematical Physics 6 (1999), 247-281.
PdfPDF (this site)
▼ Abstract
▲ Abstract The paper is devoted to four spectral problems for the Lamé system of linear elasticity in domains of R3 with compact connected boundary S. The frequency is fixed in the upper closed half-plane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are investigated: 1) S is infinitely smooth; 2) S is Lipschitz.
M Levitin, Dirichlet and Neumann heat invariants for Euclidean balls, Differential Geometry and Applications 8 (1998), 35-46.
doidoi: 10.1016/S0926-2245(97)00016-8
▼ Abstract
▲ Abstract Using the theory of heat invariants we present an efficient and economical method of obtaining the higher coefficients of the asymptotic expansion of the trace of the heat semigroup for the Dirichlet and (generalized) Neumann Laplacians acting on an m-dimensional ball. The results are presented in the form of explicit formulae for the first 10 coefficients as functions of m.
M Levitin, Fourier Tauberian theorems, Appendix in the monograph by Yu Safarov and D Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs Series, vol. 55, American Mathematical Society, Providence, R. I. (1997), 297-305.
PdfPDF (this site)
▼ Abstract
▲ Abstract The objective of this appendix is to formulate and prove Fourier Tauberian theorems as theorems of classical analysis without any reference to partial differential equations, spectral theory etc. The notion of a Tauberian theorem covers a wide range of different mathematical results. These results have the following in common. Suppose that we have some mathematical object with highly irregular behaviour (say, a discontinuous function or a divergent series) and suppose that we apply some averaging procedure which makes our object substantially more regular (say, a transformation which turns our discontinuous function into an infinitely smooth one or makes our divergent series absolutely convergent). A Tauberian theorem in our understanding is a mathematical result which recovers properties of the original irregular object from the properties of the averaged object. Tauberian theorems described in this appendix are associated mainly with the Fourier transform.
M van den Berg and M Levitin, Functions of Weierstrass type and spectral asymptotics for iterated sets, Quarterly Journal of Mathematics Oxford (2) 47 (1996), 493-509.
doidoi: 10.1093/qmath/47.4.493
▼ Abstract
▲ Abstract We construct sharp asymptotic expansions for functions of Weierstrass type. We use the results to explain the underlying phenomena in spectral asymptotics for itereated sets.
M Levitin and D Vassiliev, Vibrations of shells contacting fluid: asymptotic analysis, Acoustic Interaction with Submerged Elastic Structures, eds. A Guran, J Ripoche and F Ziegler (Series on Stability, Vibration and Control of Systems, Series B: vol 5), World Scientific, Singapore, Part 1 (1996), 310-332.
PdfPDF (this site)
doidoi: 10.1142/9789812830593_0010
▼ Abstract
▲ Abstract In this review paper, we consider free and forced harmonic vibrations of a thin elastic shell filled with or immersed into fluid. We construct the asymptotics of the eigenfrequencies and scattering frequencies in the problems of free vibrations, and of the solutions of non-homogeneous problems, using the relative shell thickness as the main asymptotic parameter.
M Levitin and D Vassiliev, Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals, Proceedings of the London Mathematical Society (3) 72 (1996), 178-214.
doidoi: 10.1112/plms/s3-72.1.188
▼ Abstract
▲ Abstract We consider the asymptotic behaviour of the volume of the Minkowski sausage, the counting function of the Dirichlet Laplacian, the partition function and the heat content for an iterated set with fractal boundary. We show, using the renewal theory (well known in probability) that in all cases the asymptotic behaviour depends essentially on whether the set of logarithms of the similitudes used in the construction of the iterated set is arithmetic.
M Levitin and D Vassiliev, Some examples of two-term spectral asymptotics for sets with fractal boundary, Operator Theory: Advances and Applications 78, Birkhäuser, Basel (1995), 227-233.
doidoi: 10.1007/978-3-0348-9092-2_25
▼ Abstract
▲ Abstract We construct a multiparametric family of sets in a Euclidean space which have fractional Minkowski dimension of the boundary. Using the renewal theory, we explicitly construct two-term spectral asymptotics for these sets.
J Fleckinger, M Levitin and D Vassiliev, Heat equation on the triadic von Koch snowflake: asymptotic and numerical analysis, Proceedings of the London Mathematical Society (3) 71 (1995), 372-396.
doidoi: 10.1112/plms/s3-71.2.372
▼ Abstract
▲ Abstract We obtain full small time asymptotic expansions of the heat content and the partition function (trace of the heat semigroup) for the triadic von Koch snowflake domain. Some numerical results are also presented.
M Levitin, Exterior spectral problem for the Douglis-Nirenberg elliptic dissipative operator, in Spectral Analysis of Complex Structures, ed. E Sanchez-Palencia, Hermann, Paris (1995), 59-69.
▼ Abstract
▲ Abstract We consider the spectral problem in an exterior domain for the special matrix differential operator appearing, for example, in the theory of vibrations of a viscous compressible fluid. We find the comntinuous spectrum, and show that the problem is in some sense similar to the classical Helmholtz equation. We also prove the existence and uniqueness of the solution of the non-homogeneous problem in the physically interesting case of real frequencies.
D V Evans, M Levitin and D Vassiliev, Existence theorems for trapped modes, Journal of Fluid Mechanics 261 (1994), 21-31.
doidoi: 10.1017/S0022112094000236
▼ Abstract
▲ Abstract A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, which corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleigh quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.
M Levitin, Vibrations of a viscous compressible fluid in bounded domains: spectral properties and asymptotics, Asymptotic Analysis 7 (1993), 15-35.
doidoi: 10.3233/ASY-1993-7103
▼ Abstract
▲ Abstract We consider the equations of small (acoustic) vibrations of a viscous compressible barotropic fluid in a bounded smooth domain under various boundary conditions. We invetigate the structure of the spectrum of the corresponding non-self-adjoint Douglis-Nirenberg elliptic system; obtain the estimates on the norm of the resolvent (in the problems of forced vibrations); constract infinite asymptotic expansions for eigenfrequencies and eigenfunctions in terms of a vanishing viscosity coefficient and determine analytically the leading terms of these expansions.
M Levitin, On a spectrum of a generalized Cosserat problem, Comptes Rendus Acad. Sci. Paris Série I 315 (1992), 925-930.
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▼ Abstract
▲ Abstract We study the spectrum of the Dirichlet problem for the second order operator pencil which coefficients are given rational meromorphic functions of the spectral parameter. Our problem generalises the classical Cosserat problem and possesses a number of applications. For a bounded and unbounded domain and in the case of the whole space we find the essential spectrum and obtain some estimates for eigenvalues.
M Levitin, Vibrations of a viscous compressible fluid in bounded and unbounded domains, Mathematical Methods in Fluid Mechanics (Lisbon, 1991), Edited by J. F. Rodrigues and A. Sequeira. Pitman Research Notes in Mathematics Series 274, Longman Scientific & Technical, Harlow, 251-255.
D Vassiliev, M Levitin, and V Lidskii, Forced oscillations of a shell immersed in a viscous compressible fluid, Funktsionalnyi Analiz i Prilozheniya 25:4 (1991), 93-95 (Russian); translation in Functional Analysis and Applications 25:4 (1991), 309-311.
doidoi: 10.1007/BF01080093
D Beilin, M Levitin, and V Polyakov, Interaction of a truncated spherical shell with an air stream parallel to the base, Structural Mechanics and Analysis of Constructions 33: 1 (1991), 17-23 (in Russian).
A Laptev, M Levitin, and D Vassiliev, A class of nonlinear variational problems arising in the theory of magnetoelasticity of thin superconducting shells, Nonlinearity 4 (1991), 821-833.
doidoi: 10.1088/0951-7715/4/3/009
▼ Abstract
▲ Abstract The authors consider the variational problem describing the static deformation of a thin elastic superconducting shell in a magnetic field; the shell is supposed to be clamped along the edge. This problem is essentially nonlinear because the functional in the problem depends on the unknown deformed shell middle surface. For sufficiently weak fields and under some additional simplifications they prove that the solution of this problem exists and is unique.
D Vassiliev, M Levitin, and V Lidskii, Forced oscillations of a thin elastic shell that is filled with a viscous compressible fluid, Doklady AN SSSR 305:2 (1989), 329-332 (in Russian); English translation in Soviet Physics Doklady 34:3 (1989), 215-217.
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M Levitin, On the spectrum of natural oscillations of a shell filled with a viscous compressible fluid, Doklady AN SSSR 295:6 (1987), 1355-1358 (in Russian); English translation in Soviet Physics Doklady 32:8 (1987), 638-639.
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M Levitin, Free vibrations of a shell filled with a viscous fluid, Mathematical Methods of Control and Data-Processing, Moscow Institute of Physics and Technology Publishing House, Moscow (1986), 132-135 (in Russian).
M Levitin and A Sudakov, Eigenvalues of a problem on the semi-axis with constant coefficients, Mathematical Methods of Control and Data-Processing, Moscow Institute of Physics and Technology Publishing House, Moscow (1985), 48-53 (in Russian).