The asymptotical behaviour of embedded eigenvalues for perturbed periodic operators

TL;DR

This paper investigates how embedded eigenvalues of perturbed periodic operators behave asymptotically near spectral boundaries, under specific decay conditions on the perturbation.

Contribution

It establishes the asymptotic behavior of embedded eigenvalues approaching spectral boundaries for perturbed periodic operators under certain decay conditions.

Findings

Embedded eigenvalues approach spectral boundaries asymptotically.

Asymptotic behavior is characterized under decay conditions on the perturbation.

Results apply to both continuous and discrete periodic operators.

Abstract

Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[\alpha_l,\beta_l]$. Under the assumption that $\limsup_{x\to \infty} x|V(x)|<\infty$ ($\limsup_{n\to \infty} n|V(n)|<\infty$), the asymptotical behaviour of embedded eigenvalues approaching to the spectral boundary is established.