Spectral analysis of the half-line Kronig-Penney model with Wigner-von Neumann perturbations

TL;DR

This paper investigates how adding oscillating, slowly decaying Wigner-von Neumann type perturbations to the half-line Kronig-Penney model preserves the band-gap structure of the spectrum while introducing critical points with potential embedded eigenvalues.

Contribution

It demonstrates that the band-gap spectral structure remains intact under Wigner-von Neumann perturbations and characterizes the emergence of critical points and embedded eigenvalues.

Findings

The absolutely continuous spectrum retains its band-gap structure.

Exactly two critical points appear in each spectral band.

Embedded eigenvalues may occur at critical points.

Abstract

The spectrum of the self-adjoint Schr\"odinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points "instable" embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains discrete.