Related papers: On Sesquilinear Forms for Lower Semibounded (Singu…

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends bilinearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary…

Let $T$ be an o-minimal theory extending the theory of real closed ordered fields. An $H_T$-field is a model $K$ of $T$ equipped with a $T$-derivation such that the underlying ordered differential field of $K$ is an $H$-field. We study…

In this paper we study a Sturm--Liouville operator $Ly=-y''+q(x)y$ in the space $L_2[0,\pi]$ with Direchlet boundary conditions. Here the potential $q$ is a fitst order distribution $q\in W_2^{-1}[0,\pi]$. Such operators were defined in our…

We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth boundary. Each of these quadratic forms specifies a semi-bounded…

The semiclassical limit of the FZZT Liouville theory on the upper half plane with bulk operators of arbitrary type and with elliptic boundary operators is analyzed. We prove the Polyakov conjecture for an appropriate classical Liouville…

This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm-Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced…

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (-…

We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class…

We consider a Sturm-Liouville operator a with integrable potential $q$ on the unit interval $I=[0,1]$. We consider a Schr\"odinger operator with a real compactly supported potential on the half line and on the line, where this potential…

We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local…

We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operator \[ A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx} \] acting in $L^2_{w}(-b,b)$. It is assumed that $w,r\in L^1_{\loc}(-b,b)$ are even and…

In this paper, further to the point interaction method for inverse Sturm-Liouville problems on finite intervals firstly proposed in our previous work, we will continue to generalize this method to the inverse eigenvalue problems for…

In this paper, we investigate the sampling analysis associated with discontinuous Sturm-Liouville problem which has transmission conditions at two points of discontinuity also contains an eigenparameter in a boundary condition and two…

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,…

All self-adjoint extensions of minimal linear relation associated with the discrete symplectic system are characterized. Especially, for the scalar case on a finite discrete interval some equivalent forms and the uniqueness of the given…

Under appropriate assumptions, we show that all bounded entire solutions to a class of semilinear elliptic systems are confined in a convex domain. Moreover, we prove a Liouville type theorem in the case where the domain is strictly convex.…

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is…

In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: \begin{equation*} x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi) \end{equation*}…

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as…

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for…