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.

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered. MSC-cl...

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem second-order differential equation containing fractional derivative operator. The differentiation operator was considered according to two definitions: Gerasimov-Caputo Riemann-Liouville-Visualizations eigenfunctions, biorthogonal system, distribution on real axis were present...

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∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular cri...

The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the di...

The method proposed here has been devised for solution of the spectral problem for the Lamé wave equation (see [2]), but extended lately to more general problems. This method is based on the phase function concept or the Prüfer angle determined by the Prüfer transformation cot θ(x) = y′(x)/y(x), where y(x) is a solution of a second order self-adjoint o.d.e. The Prüfer angle θ(x) has some useful...

We consider eigenvalue problems for self-adjoint Sturm-Liouville difference equations of any even order. It is well known that such problems with Dirichlet boundary conditions can be transformed into an algebraic eigenvalue problem for a banded, real-symmetric matrix, and vice versa. In this article it is shown that such a transform exists for general separated, self-adjoint boundary conditions...