Ramanujan's master theorem for Sturm-Liouville operator

Self-similar functions in $L_2[0,1]$ and Sturm--Liouville Problem with singular indefinite weight

In the paper the Sturm-Liouville problem $-y''-ρy=0$, $y(0)=y(1)=0$ is studied. $ρ$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained.

Indefinite Sturm--Liouville problem for some classes of self-similar singular weights

We study the Sturm-Liouville problem $-y''-ρy=0$, $y(0)=y(1)=0$. $ρ$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained. In this paper we consider two cases of self-similarity of $P$: 1) arithmetic degenerate, and 2) non-arithmetic.

Sturm-Liouville operators with matrix distributional coefficients

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.

基于 Sturm-Liouville 理论的圆盘上卷积的离散谱

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。

Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm--Liouville equations

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.

Characterization of the spectrum of irregular boundary value problem for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.

Asymptotics of eigenvalues of Sturm--Liouville problem with discrete self-similar weight

The Sturm--Liouville problem $-y''-λρy=0$, $y(0)=y(1)=0$, where $ρ$ is a generalized derivative of self-similar function $P\in L_2[0,1]$ with spectral degree D=0, is studied. Asymptotic formulas for eigenvalues are obtained.

Convergence of regular approximations to the spectra of singular fourth order Sturm-Liouville problems

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth order Sturm-Liouville problem near a singular endpoint. Of particular interest are the results in the so called lim-3 case, which has no analogue in second order singular problems.

On periodic boundary value problem for the Sturm-Liouville operator

We consider the Sturm-Liouville operator Lu=u''-q(x)u with periodic or antiperiodic boundary conditions. It is shown that depending of Fourier coefficients of the potential q(x) the system of root functions may have or may not have the basis property.

Inverse problem of the spectral analysis for the Sturm-Liouville operator with non-separated boundary conditions and spectral parameter in the boundary condition

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.

Random Sturm Liouville Operators

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

A method of calculation of eigenvalues of the problem $-y''-λρy=0$, $y(0)=y(1)=0$, where $ρ$ is a distribution having self-similar primitive $P\in L_2[0,1]$, is described.

Inverse spectral problems for Sturm--Liouville operators with matrix-valued potentials

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.

Inverse spectral problems for energy-dependent Sturm-Liouville equations

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.

About one inverse problem for the Sturm-Liouville operator

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.

自伴Sturm-Liouville算子的下界

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。

Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $ω$-parametric analytic family better than order of $(ω\lnω)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $ω^{-m}$.

Resolvent convergence of Sturm-Liouville operators with singular potentials

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.