Nataliya Pronska
We study asymptotics of eigenvalues, eigenfunctions and norming constants of singular energy-dependent Sturm--Liouville equations with complex-valued potentials. The analysis essentially exploits the integral representation of solutions, which we derive using the connection of the problem under study and a Dirac system of a special form.
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Characterization of the spectrum of irregular boundary value problem for the Sturm-Liouville operator
本文研究了具有任意复值位势q（x）和不规则边界条件的Sturm-Liouville方程的谱问题。我们建立了一个复数集是这类算子的谱的充要条件。
Asymptotics of eigenvalues of Sturm--Liouville problem with discrete self-similar weight
研究了Sturm-Liouville问题$-y''-λρy=0$，$y(0)=y(1)=0$，其中$ρ$是谱度D=0的自相似函数$P\in L_2[0,1]$的广义导数.得到了特征值的渐近公式。
Convergence of regular approximations to the spectra of singular fourth order Sturm-Liouville problems
我们证明了一些新的结果，证明了使用区间截断作为一种手段正则化奇异四阶Sturm-Liouville问题附近的奇异端点。特别感兴趣的是在所谓的lim-3的情况下，没有类似的二阶奇异问题的结果。
On periodic boundary value problem for the Sturm-Liouville operator
考虑Sturm-Liouville算子Lu= u“-q（x）u具有周期或反周期边界条件.证明了根函数系可能具有或不具有基性质，取决于位势q（x）的傅里叶系数.
Selfadjoint Sturm-Liouville operators $H_ω$ on $L_2(a,b)$ with random potentials are considered and it is proven, using positivity conditions, that for almost every $ω$ the operator $H_ω$ does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as $H_ω$ but in $L_2(\tilde a,\tilde b)$ where $(\tilde a,\tilde b)\subset(a,b)$.
On calculation of eigenvalues of Sturm-Liouville problem with indefinite weight having self-similar primitive
描述了计算问题$-y''-λρy=0$，$y(0)=y(1)=0$的特征值的方法，其中$ρ$是具有自相似基元$P\in L_2[0,1]$的分布。
We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.
We consider the spectral problems for the Sturm-Liouville operator generated by the Dirichlet, Neumann, Dirichlet-Neumann and Neumann-Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann-Dirichlet problems is proved. Also the necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is proved. An application to periodic and anti-periodic problems is given.
The paper investigates spectral properties of multi-interval Sturm-Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. Results of paper are new for one-interval boundary value problems as well.
We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.
In this paper we prove an analogue of Ramanujan's master theorem in the setting of Sturm Liouville operator.
Self-similar functions in $L_2[0,1]$ and Sturm--Liouville Problem with singular indefinite weight
本文研究Sturm-Liouville问题$-y''-ρy=0$，$y(0)=y(1)=0$. $ρ$是函数$P\in L_2[0,1]$的广义导数。对于自相似的$P$，得到了特征值的渐近公式。
Lower bounds for self-adjoint Sturm-Liouville operators
本研究针对加权L²希尔伯特空间中三系数Sturm-Liouville微分表达式对应的自伴算子，给出其下界的估计结果。
Indefinite Sturm--Liouville problem for some classes of self-similar singular weights
本文研究Sturm-Liouville问题$-y''-ρy=0$，$y(0)=y(1)=0$ . $ρ$是函数$P\in L_2[0,1]$的广义导数。对于自相似的$P$，得到了特征值的渐近公式。本文考虑了$P$自相似的两种情况：1）算术退化和2）非算术退化。
The paper deals with singular Sturm-Liouville expressions with matrix-valued distributional coefficients. Due to a suitable regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent convergence is investigated and all self-adjoint, maximal dissipative, and maximal accumulative extensions are described in terms of homogeneous boundary conditions of the canonical form.
Riesz bases of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions. II
考虑具有两个仿射依赖于特征参数的自伴边界条件的正则不定Sturm-Liouville问题。我们给出了这个Sturm-Liouville问题的根向量可以被选择成相应的加权希尔伯特空间的Riesz基的充分条件.
In this paper we consider the Sturm-Liuoville operator in the Hilbert space $L_2$ with the singular complex potential of $W^{-1}_2$ and two-point boundary conditions. For this operator we give sufficient conditions for norm resolvent approximation by the operators of the same class.
We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.
Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I
本文考虑一个正则不定Sturm-Liouville问题，它具有两个自伴边界条件，其中一个依赖于特征参数.我们给出了这个Sturm-Liouville问题的每个根子空间的一个基可以被选择的充分条件，使得所有这些基的并集构成相应加权希尔伯特空间的Riesz基.
This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.
In this work, we study discontinuous Sturm-Liouville type problems with eigenparameter dependent boundary condition and transmission conditions at three interior points. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We show that the eigenvalues of the problem are analytically simple, and the eigenfunctions of A are complete in H.
Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems
本文给出了Sturm-Liouville逆理论中的一些否定结果，这意味着我们不能用$ω$ -参数解析族比$(ω\lnω)^{-(m+1)}$阶更好地逼近$\mathbb{R}^+$上具有$m+1$可积导数的任何位势.其次，我们证明了Sturm-Liouville算子的特征值和特征值的一个估计以及一类积分方程解的一些性质。这使我们能够从[Henkin-Novikova]中推导出一些关于最佳重建公式的积极结果，给出了$ω^{-m}$阶的几乎最佳公式。
We investigate the discrete Sturm-Liouville equations. In the present situation, the right hand side of the equation does not give rise to a positive definite quadratic form and we use instead the left hand side to define such a form. We prove in this paper that this form determines a Hilbert space (such problems are called left-definite).
Discrete Spectra of Convolutions on Disks using Sturm-Liouville Theory
本文基于 Sturm-Liouville 理论系统研究了圆盘上函数卷积的离散谱方法，并探讨了零值边界条件和导数边界条件下的理论表现。
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