On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues

2024/10/12 by Lyu, Kang, Yang, Chuanfu
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 34L05. Secondary: 34A30 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2410.09509

Abstract

In this paper, we consider Schrödinger operators on L²(0,∞) given by Hu=(H₀+V)u=-u^′′+V₀u+Vu,where V₀ is real, 1-periodic and V is the perturbation. It is well known that under perturbations V(x)=o(1) as x→∞, the essential spectrum of H coincides with the essential spectrum of H₀. We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set S inside the essential spectrum, we can construct perturbations with S contained in the set of eigenvalues if the resonant eigenvalues in S satisfy some condition. In particular, if S is a finite set (or countable set), we can construct perturbation with V(x)=(O(1))/(x) (or \absV(x)≤(h(x))/(1+x)) as x→∞ if the resonant eigenvalues of S appear in the same spectral bands or large separate spectral bands, where h(x) is any given function with lim_x→∞h(x)=∞.

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