The eigenvalues of Sturm-Liouville (SL) problems depend not only continuously but smoothly on the problem. An expression for the derivative of the n-th eigenvalue with respect to a given parameter: an endpoint, a boundary condition constant, a coefficient or weight function, is found.

This paper deals with the boundary value problem involving the differential equation ell y:=-y''+qy=lambda y, subject to the eigenparameter dependent boundary conditions along with the following discontinuity conditions y(d+0)=a y(d-0), y'(d+0)=ay'(d-0)+b y(d-0). In this problem q(x), d, a , b are real, qin L^2(0,pi), din(0,pi) and lambda is a parameter independent of x. By defining a new...

Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem certain type an eigenparameter-dependent condition, this specified condition was constructed. It has been proven that each is equivalent to given problem.

We study finite difference approximations of solutions of direct and inverse Sturm–Liouville problems, in a finite or infinite interval on the real line. The discretization is done on optimal grids, with a three-point finite difference stencil. The optimal location of the grid points is calculated via a rational approximation of the Neumann-to-Dirichletmap and the latter converges exponentially...

In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin s XXZ model of a finite size and the q-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz equations give rise to the polynomial solutions of a second order q-difference equation in terms of Askey-Wilson operator. The more general form of Bethe Ansatz e...

for some λ ∈ C, x ∈ I = [a, b], and y ∈ C2(I). It was first introduced in a 1837 publication [7] by the eminent French mathematicians Joseph Liouville (1809 1882) and Jacques Charles François Sturm (1803 1855). At this point, our initial questions might be: why is this problem important? What can we say about the structure of this equation? How about the solutions? We will tackle a few basic re...

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

We study the nonlinear boundary value problem consisting of the equation y+w(t)f(y) = 0 on [a, b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a twopoint separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different...

For any positive integer n and any given n distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of n numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.