Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

On the spectral theory of singular indefinite Sturm-Liouville operators

We consider a singular Sturm-Liouville differential expression with an indefinite weight function and we show that the corresponding self-adjoint differential operator in a Krein space locally has the same spectral properties as a definitizable operator.     

Spectral asymptotics for Sturm-Liouville equations with indefinite weight

The Sturm-Liouville equation

is considered subject to the boundary conditions

We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.

     

Regular singular Sturm-Liouville operators and their zeta-determinants

We consider Sturm-Liouville operators on the line segment [0,1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.     

Sturm-liouville operators with singular potentials

This paper deals with Sturm-Liouville operators generated on a finite interval and on the whole axis by the differential expressionl(y)=−y ^" +q(x)y, whereq(x) is a distribution of first order, such that . The minimal and maximal operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint extensions of the minimal operator are described and the asymptotics of the eigenvalues of these extensions is found. It is proved that the constructed operator coincides with the norm resolvent limit of the Sturm-Liouville operators generated by smooth potentialsq _n , provided that the condition holds. The convergence of the spectra of these operators to the spectrum of the limit operator is also proved. Similar results are obtained in the case of the whole axis. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 897–912, December, 1999.     

边界条件含特征参数的Sturm-Liouville问题的特征值比率

给出修正Prfer变换的几何意义以及相关性质,并利用上述性质讨论Sturm-Liouville问题-(py′)′+gy=λωy,x∈[0,1],参数边界条件为y(0)=0与((py′)(1))/(y(1))=aλ+b,得到了当a>0,q≥0的特征值比率上界,这把Ashbaugh等人的结果推广到边界条件含参数的情形.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

L~1-uniqueness of Sturm-Liouville operators

In this paper, we give two sufficient and necessary conditions for the L1-uniqueness of one-dimensional Sturm-Liouville operator Lf = af″ + bf′- V f, f∈D = C 0 ∞ (I), where I is an open interval of R and V≥0.     

Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields

We consider the Schrödinger operator with magnetic field,

Assuming that and is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.

     

Sturm-Liouville operators in the half axis with local perturbations

We give conditions which imply equivalence of the Lebesgue measure with respect to a measure μ generated as an average of spectral measures corresponding to Sturm-Liouville operators in the half axis. We apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Stieltjes Sturm-Liouville equations: Eigenvalue dependence on problem parameters

We examine properties of eigenvalues and solutions to a 2n-dimensional Stieltjes Sturm-Liouville eigenvalue problem. Existence and uniqueness of a solution has been established previously. An earlier paper considered the corresponding initial value problem and established conditions which guarantee that solutions depend continuously on the coefficients [L.E. Battle, Solution dependence on problem parameters for initial value problems associated with the Stieltjes Sturm-Liouville equations, Electron. J. Differential Equations 2005 (2) (2005) 1-18]. Here, we find conditions which guarantee that the eigenvalues and solutions depend continuously on the coefficients, endpoints, and boundary data. For a simplified two-dimensional problem, we find conditions which guarantee the eigenvalues to be differentiable functions of the problem data.     

A singular Sturm-Liouville equation under homogeneous boundary conditions

Given α>0 and f∈L²(0,1), we are interested in the equation

     

Sharp eigenvalue asymptotics for fourth order operators on the circle

We determine high energy asymptotics of eigenvalues of fourth order operator on the circle.