Related papers: On Sturm-Liouville Equations with Several Spectral…

Exact solutions to the d-dimensional Schroedinger equation, d\geq 2, for Coulomb plus harmonic oscillator potentials V(r)=-a/r+br^2, b>0 and a\ne 0 are obtained. The potential V(r) is considered both in all space, and under the condition of…

We study the Sturm-Liouville problem $-y''-\rho y=0$, $y(0)=y(1)=0$. $\rho$ 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…

In this paper we consider the following Sturm-Liouville equation \[ \left\{ \begin{aligned} -(x^{2\alpha}u'(x))'+u(x)&=f(x) && \text{in } (0,1],\\ u(1)&=0 \end{aligned} \right. \] where $\alpha<1$ is a nonzero real number and $f$ belongs to…

By using a generalization of Sturm-Liouville problems in $q$-difference spaces, a class of symmetric $q$-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order…

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the \textit{regularized sampling method}. A few numerical examples among which singular…

We introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the…

We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this…

We study the problem of correct solvability in the space $L_p(\mathbb R),$ $p\in[1,\infty)$ of the equation $$ -(r(x) y'(x))'+q(x)y(x)=f(x),\quad x\in \mathbb R $$ under the conditions $$r>0,\quad q\ge 0,\quad \frac{1}{r}\in L_1(\mathbb…

Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y" + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi…

We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues…

In this paper, we for the first time get constructive solution for the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of…

In this paper, we consider a wave equation on a bounded domain with a Sturm-Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is…

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations…

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

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary…

The paper deals with two inverse problems for Sturm--Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\,…

In this paper we study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from two spectra. We give a reconstruction algorithm and establish existence and uniqueness of reconstruction. Our approach…

Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these…

This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a…

In this paper we study some solution techniques of differential-difference equation $$ y'(x) = y(x + 1/2)- y(x- 1/2),$$ first without an initial condition and then with some initial function $h$ defined on the unit interval $ [-1/2, 1/2]$.…