Papers and Preprints

Where available, links to the final published texts are provided; archived preprints (also linked) may differ from final versions. Hovering the mouse over the word "Abstract" should show the abstract of the paper on the right top (doesn't work in IE).

Rafael Benguria, M Levitin, and L Parnovski, Fourier transform, null variety, and Laplacian's eigenvalues, preprint (2008).
PdfPDF (arXiv:math)
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.
L Boulton, M Levitin, and M Marletta, A PT-symmetric periodic problem with boundary and interior singularities, preprint (2007).
PdfPDF (arXiv:math)
Abstract We prove that the spectrum of a certain PT-symmetric periodic problem is purely real. Our results extend to a larger class of potentials those recently found by Brian Davies and John Weir.
M Levitin and M Marletta, A simple method of calculating eigenvalues and resonances in domains with infinite regular ends, Proc. Royal Soc. Edinburgh, 138A (2008), 1043-1065.
PdfPDF (arXiv:math)
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, Math. Nachrichten, 281 (2008), 272-281.
PdfPDF (arXiv:math)
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 Schroedinger operators, J. Phys. A: Math. Gen., 40 (2007), 9319-9329.
PdfPDF (arXiv:math)
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.
M Levitin, L Parnovski, and I Polterovich, Isospectral domains with mixed boundary conditions, J. Phys. A: Math. Gen., 39 (2006), 2073-2082.
PdfPDF (arXiv:math)
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, J. Comp. and Appl. Math. 194 (2006), 141-155.
PdfPDF (arXiv:math)
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?, Int. Math. Res. Notices 2005:63 (2005), 3967-3985.
PdfPDF (arXiv:math)
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 J. Math. Anal. 37, issue 5 (2006),1465-1481.
PdfPDF (arXiv:math)
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 J. Numer. Analysis 24 (2004), 393-416.
PdfPDF (arXiv:math).
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 J. Comput. Math. 6 (2003), 1-17.
PdfPDF (arXiv:math)
AbstractWe 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)
AbstractWe 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, J. Functional Analysis 192 (2002), 425-445.
PdfPDF (arXiv:math)
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)
M S Agranovich, B A Amosov and M Levitin, Spectral problems for the Lamé system with spectral parameter in boundary bonditions on smooth or nonsmooth boundary, Russian Jour. Math. Physics 6 (1999), 247-281.
PdfPDF (this site)
AbstractThe 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 Applicat. 8 (1998), 35-46.
AbstractUsing 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 , Series Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, R. I. (1997), 297-305.
PdfPDF (this site)
AbstractThe 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, Quarter. J. Math. Oxford (2) 47 (1996), 493-509.
AbstractWe 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)
AbstractIn 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, Proc. London Math. Soc. (3) 72 (1996), 178-214.
AbstractWe 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.
AbstractWe 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, Proc. London Math. Soc. (3) 71 (1995), 372-396.
AbstractWe 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, Spectral Analysis of Complex Structures, ed. E Sanchez-Palencia, Hermann, Paris (1995), 59-69.
AbstractWe 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, J. Fluid Mech. 261 (1994), 21-31.
AbstractA 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.
AbstractWe 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, C. R. Acad. Sci. Paris Série I 315 (1992), 925-930.
AbstractWe 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, Funktsional. Anal. i Prilozhen. 25 (1991), no. 4, 93-95 (Russian); translation in Funct. Anal. Appl. 25 (1991), no. 4, 309-311.
D Beilin, M Levitin, and V Polyakov, Interaction of a truncated spherical shell with an air stream parallel to the base , Building Mechanics and Analysis of Constructions 33, no. 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.
AbstractThe 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, no. 2 (1989), 329-332 (in Russian); English translation in Soviet Physics Doklady 34, no. 3 (1989), 215-217.
M Levitin, On the spectrum of natural oscillations of a shell filled with a viscous compressible fluid, Doklady AN SSSR 295, no. 6 (1987), 1355-1358 (in Russian); English translation in Soviet Physics Doklady 32, no. 8 (1987), 638-639.
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).