The one dimensional infinite square well with variable mass

We introduce a numerical method to obtain approximate eigenvalues for some problems of Sturm-Liouville type. As an application, we consider an infinite square well in one dimension in which the mass is a function of the position. Two situations are studied, one in which the mass is a differentiable function of the position depending on a parameter $b$. In the second one the mass is constant except for a discontinuity at some point. When the parameter $b$ goes to infinity, the function of the mass converges to the situation described in the second case. One shows that the energy levels vary very slowly with $b$ and that in the limit as $b$ goes to infinity, we recover the energy levels for the second situation.

On the applications of Hardy class functions in scattering theory

This paper is a response to an article (R. de la Madrid, Journal of Physics A: Mathematical and General, 39,9255-9268 (2006)) recently published in Journal of Physics A: Mathematical and Theoretical. The article claims that the theory of resonances and decaying states based on certain rigged Hilbert spaces of Hardy functions is physically untenable. In this paper we show that all of the key conclusions of the cited article are the result of either the errors in mathematical reasoning or an inadequate understanding of the literature on the subject.

Riggings of locally compact abelian groups

We obtain a set of generalized eigenvectors that provides a generalized spectral decomposition for a given unitary representation of a commutative, locally compact topological group. These generalized eigenvectors are functionals belonging to the dual space of a rigging on the space of square integrable functions on the character group. These riggings are obtained through suitable spectral measure spaces.