In this paper, an inverse nodal problem for a second-order differential equation having a chemical potential on a finite interval is investigated. First, we estimate the nodal points and nodal lengths of differential operator. Then, we show that the potential can be uniquely determined by a dense set of nodes of the eigenfunctions.

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the...

We study the division of components of energy eigenfunctions, as the expansion of perturbed states in unperturbed states, into nonperturbative and perturbative parts in a three-orbital schematic shell model possessing a chaotic classical limit, the Hamiltonian of which is composed of a Hamiltonian of noninteracting particles and a perturbation. The perturbative parts of eigenfunctions are expan...

Abstract We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. first prove existence eigenfunctions. then set up a framework for to discretize equation defining eigenfunctions, obtain numerical approximations eigenvalue eigenfunction using finite element method. Finally, we show that conve...

The class of differential-equation eigenvalue problems −y′′(x)+x2N+2y(x) = xNEy(x) (N = −1, 0, 1, 2, 3, . . .) on the interval −∞ < x < ∞ can be solved in closed form for all the eigenvalues E and the corresponding eigenfunctions y(x). The eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions. The eigenfunctions can be rewritten as products of polynomial...

In this paper we investigate a Sturm–Liouville eigenvalue problem on time scales. Existence of the eigenvalues and eigenfunctions is proved. Mean square convergent and uniformly convergent expansions in the eigenfunctions are established. AMS subject classification: 34L10.

We present results on special eigenfunctions for differences of elliptic CalogeroMoser type Hamiltonians. We show that these results have a bearing on the existence of joint Hilbert space eigenfunctions for the commuting Hamiltonians.