Abstract:
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.

Abstract:
A class of singular semipositone fourth order differential equations with Sturm-Liouville boundary value conditions is considered. Some new results on the existence of $C^2$ positive solutions and $C^3$ positive solutions for this class of differential equations are derived. Finally, some relations between the solutions and the Green's functions are derived.

Abstract:
We discuss direct and inverse spectral theory of self-adjoint Sturm-Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl-Titchmarsh theory for these relations. Consequently, we apply de Branges' subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa-Holm equation.

Abstract:
Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization corresponding operators are correctly defined as quasi-differential. Their resolvent approximation is investigated and all self-adjoint and maximal dissipative extensions and generalized resolvents are described in terms of homogeneous boundary conditions of the canonic form. Some results are new for the case $p(t)\equiv 1$ as well.

Abstract:
We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz and Poschel.

Abstract:
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 known that the $\zeta$-function of this operator $\zeta_L(s)=\sum_{\lambda\in\spec(L)\setminus\{0\}} \lambda^{-s}$ has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to Ray and Singer the $\zeta$-regularized determinant of $L$ is defined by $\detz(L):=\exp(-\zeta_L'(0)).$ In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation $Ly=0$ generalizing earlier work of S. Levit and U. Smilansky, T. Dreyfus and H. Dym, and D. Burghelea, L. Friedlander and T. Kappeler. More precisely we prove the formula $\detz(L)=\frac{\pi W(\psi,\phi)} {2^{\nu_0+\nu_1} \Gamma(\nu_0+1)\Gamma(\nu_1+1)}.$ Here $\phi, \psi$ is a certain fundamental system of solutions for the homogeneous equation $Ly=0$, $W(\phi, \psi)$ denotes their Wronski determinant, and $\nu_0, \nu_1$ are numbers related to the characteristic roots of the regular singular points $0, 1$.

Abstract:
Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with Sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential Sinc collocation method clearly illustrate the advantage of using the double exponential formula.

Abstract:
Sturm-Liouville spectral problem for equation $-(y'/r)'+qy=\lambda py$ with generalized functions $r\ge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $r\equiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.

Abstract:
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.

Abstract:
We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh $m$-functions. Also we obtain necessary conditions for regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator $-\frac{(\sgn x)}{(3|x|+1)^{-4/3}} \frac{d^2}{dx^2}$ acting in the Hilbert space $L^2(\R, (3|x|+1)^{-4/3}dx)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type $(\sgn x)(-d^2/dx^2+q(x))$ with the same properties.