On the spectrum of the Dirichlet Laplacian in a narrow strip

We consider the Dirichlet Laplacian Δ_∈ in a family of bounded domains {−a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ_∈.     

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy bòB²{\beta \int B^{2}} and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h ² ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h^1+e{h^{1+\varepsilon}} , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H ^s , the simplest L ₂-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues.     

Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

We consider quite general h-pseudodifferential operators on R ⁿ with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.     

Asymptotics for the voltage potential in a periodic network with localized defects

We compare the potential u_h of a 2D regular lattice of conductors of size h, to the potential u_h,d of a defective lattice, where some conductive links have different conductivities. We show that to first order in h, each defect contributes to the difference u_h,d?u_h as a product of three terms: A polarization matrix, the gradient of the potential u of the limiting continuous medium obtained as h → 0, and the gradient of Green's function of the limiting medium. Establishing the asymptotics of u_h,d?u_h involves uniform W^1,∞ estimates on the potentials u_h. Copyright © 2007 John Wiley & Sons, Ltd.     

On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator L\mathcal{L}. It can be seen in this paper that the auxiliary parameter (^h/_2p),\hbar, which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.     

A degenerate kernel method for eigenvalue problems of compact integral operators

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by a degenerate kernel method. By interpolating the kernel of the integral operator in both the variables, we prove that the error bounds for eigenvalues and for the distance between the spectral subspaces are of the orders h ^2r and h ^r respectively. By iterating the eigenfunctions we show that the error bounds for eigenfunctions are of the orders h ^2r . We give the numerical results.      

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weightedL ²-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.     

Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

A finite-difference approximation for the eigenvalues of the clamped plate

Summary A finite-difference scheme is given for the eigenproblem of the clamped plate. The discrete eigenvalues and eigenvectors are shown to converge to the continuous eigenvalues and eigenvectors likeO(h ²) andO (h ² logh 1/2) respectively.This work, supported by the U.S. Department of the Navy under Contract N 00017-62-C-0604.     

Exact <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L₂-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.Partially supported by grants of RFBR 01-01-00245 and 02-01-01099.     

A Hopf bifurcation theorem for difference equations approximating a differential equation

If an ordinary differential equation is discretizised near an asymptotically stable stationary solution with a pair of imaginary eigenvalues by Euler's method with constant step lengthh, small invariant attracting cycles of radiusO(h ^1/2) will appear. This Hopf bifurcation theorem is applied to prove the existence of limit cycles in certain difference equations occurring in biomathematics (hypercycle, two loci-two alleles) and is also extended to general Runge—Kutta methods.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of Two Linearized Problems Modeling Viscous Two‐Layer Flows

Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip‐like domain Π = {x = (x₁, x₂) ∈ ℝ² : x₁ ∈ ℝ¹, (0 < x₂ < h_*) ∨ (h_* < x₂ < 1)}. The first problem arises in the linearization of a two‐layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two‐layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π. The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.     

Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners

The Neumann realization for the Schr?dinger operator with magnetic field is considered in a bounded two-dimensional domain with corners. This operator is associated with a small semi-classical parameter h or, equivalently, with a large magnetic field.We investigate the behavior of its eigenpairs as h tends to zero, like in a semi-classical limit. We prove, in the situation where the domain is a polygon and the magnetic field is constant, that the lowest eigenvalues are exponentially close to those of model problems associated with the corners. We approximate the corresponding eigenvectors by linear combinations of functions concentrated in corners at the scale If the domain has curved sides and the magnetic field is smoothly varying, we exhibit a full asymptotics for eigenpairs in powers of Communicated by Christian Gérard Submitted: October 13, 2005 Accepted: December 19, 2005