Two-term Asymptotic Formula Research Articles

Behaviour of large eigenvalues for the asymmetric quantum Rabi model

We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m − ) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O ( m − 1 / 4 ), when m tends to infinity.

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

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.

Asymptotic derivation of refined dynamic equations for a thin elastic annulus

Low-frequency vibrations of a thin elastic annulus are considered. The dynamic equations of plane strain are subjected to asymptotic treatment beyond the leading-order approximation. The main peculiarity of the considered problem is a specific degeneration associated with the effect of the almost inextensible midline of the annulus, resulting in a few unexpected features of the mechanical behaviour. In particular, it is discovered that the leading-order even component of the circumferential stress is not uniform across the thickness, as is usually assumed, and can be determined only at the next order. The derived refined equations also govern vibrations of a cylindrical shell at the lowest cut-off frequencies. The two-term asymptotic formula obtained for the latter fully agrees with the expansion of the transcendental dispersion relation for plane strain but does not coincide in the second term with the prediction of the Kirchhoff–Love theory for thin shells.

THE STEKLOV SPECTRUM OF CUBOIDS

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type

We consider a two-dimensional spectral problem of Steklov type for the Laplace operator in a domain divided into two parts by a perforated partition with a periodic microstructure. The Steklov boundary condition is imposed on the lateral sides of the perforation, the Neumann condition on the remaining part of the boundary, and the Dirichlet and Neumann conditions on the outer boundary of the domain. We construct and justify two-term asymptotic expressions for the eigenvalues of this problem. We also construct a two-term asymptotic formula for the corresponding eigenfunctions.

Curvature-induced bound states in Robin waveguides and their asymptotical properties

We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than π, and it is void if the domain is concave.

Spectral theoretic characterization of the massless Dirac operator

We consider an elliptic self-adjoint first-order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive λ. We derive an explicit two-term asymptotic formula for the spectral function as λ to + ∞, expressing the second asymptotic coefficient via the trace of the subprincipal symbol and the geometric objects encoded within the principal symbol-metric, torsion of the teleparallel connection and topological charge. We then address the question: is our operator a massless Dirac operator on half-densities? We prove that it is a massless Dirac operator on half-densities if and only if the following two conditions are satisfied at every point of the manifold: (a) the subprincipal symbol is proportional to the identity matrix; and (b) the second asymptotic coefficient of the spectral function is zero.

Sharp trace asymptotics for a class of 2D-magnetic operators

In this paper we prove a two-term asymptotic formula for for the spectral counting function for a 2D magnetic Schrodinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formulas announced by Kunz. Moreover, the same theorem provides several other results on the integrated density of states for operators of the type (−ih∇−μA)^2 in L^2(Ω) with Dirichlet boundary conditions.

Asymptotic behavior of normal derivatives of eigenfunctions for the Dirichlet Laplacian

We give answers to the problem posed by Ozawa in [S. Ozawa, Asymptotic property of eigenfunctions of the Laplacian at the boundary, Osaka J. Math. 30 (1993) 303–314]. For the Dirichlet Laplacian in a bounded domain, we define the function E ( λ , x ) from the normal derivatives, at a boundary point x, of the eigenfunctions whose corresponding eigenvalues do not exceed λ. If the domain is a ball, we show that Ozawaʼs conjecture is true, namely that E ( λ , x ) satisfies a two-term asymptotic formula as λ → ∞ . For a general C 2 bounded domain, we improve the remainder estimate in the one-term asymptotic formula for E ( λ , x ) , which Ozawa obtained.

A Note on Perturbation Formulae for the Surface-Wave Speed Due To Perturbations in Material Properties

In a recent paper by Tanuma and Man, a two-term asymptotic formula was derived for the speed of surface waves propagating in an anisotropic elastic half-space whose elastic moduli differ only slightly from those for a (base) isotropic elastic material. This formula disagrees with that derived by Delsanto and Clark in an earlier paper using a different method. In this short note, we use a simple procedure to derive another two-term asymptotic formula for the surface-wave speed. Our formula takes the same compact form even if the base material is generally anisotropic. We show that when an error in the work of Delsanto and Clark is corrected, the three different methods do give equivalent results.

SEMI-CLASSICAL ASYMPTOTICS OF THE SPECTRAL FUNCTION OF PSEUDODIFFERENTIAL OPERATORS

ABSTRACT We consider the asymptotic behaviour of the spectral function of a self-adjoint h pseudodifferential operator in the limit as h → 0 . Adapting methods developed in [1] to the semi-classical (non-homogeneous) setting, conditions are found under which a two-term asymptotic formula for the spectral function at a point on the diagonal may be written down, or under which so-called clustering of the spectral function occurs. To illustrate the results we consider the example of a Schrödinger operator with quadratic potential V.

EXACT ASYMPTOTICS OF THE SPECTRUM OF A BOUNDARY VALUE PROBLEM, AND PERIODIC BILLIARDS

A two-term asymptotic formula for the distribution function of the eigenvalues of an elliptic boundary value problem is derived. In contrast to previous work, it is not assumed that the set of points from which the periodic billiard trajectories emanate has measure zero. Bibliography: 21 titles.

On the second term of the spectral asymptotics of the transmission problem

This article is devoted to the investigation of spectral asymptotics for the second-order differential operator with transmission boundary conditions. A two-term asymptotic formula is obtained for the distribution function of eigenvalues under some assumptions not excluding the reach set of periodic billiard trajectories.