The strong form of the Levinson theorem for a distorted KP potential

TL;DR

This paper derives a modified Levinson theorem for a distorted Kronig-Penney potential, linking phase shifts at band edges to bound states, with a novel alternating sign factor, extending the theorem's applicability to quasi-periodic systems.

Contribution

It introduces a new form of the Levinson theorem for quasi-periodic potentials, incorporating an alternating sign factor and relating phase shifts to bound states in a distorted Kronig-Penney model.

Findings

Derived the strong Levinson theorem for distorted quasi-periodic potentials.

Established the relationship between phase shifts and bound states at band edges.

Presented an overall connection between phase shifts and total bound states.

Abstract

We present a heuristic derivation of the strong form of the Levinson theorem for one-dimensional quasi-periodic potentials. The particular potential chosen is a distorted Kronig-Penney model. This theorem relates the phase shifts of the states at each band edge to the number of states crossing that edge, as the system evolves from a simple periodic potential to a distorted one. By applying this relationship to the two edges of each energy band, the modified Levinson theorem for quasi-periodic potentials is derived. These two theorems differ from the usual ones for isolated potentials in non-relativistic and relativistic quantum mechanics by a crucial alternating sign factor $(-1)^{s}$, where $s$ refers to the adjacent gap or band index, as explained in the text. We also relate the total number of bound states present in each energy gap due to the distortion to the phase shifts at its edges. At the end we present an overall relationship between all of the phase shifts at the band edges and the total number of bound states present.