Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials

TL;DR

This paper investigates the discrete spectrum of Schr"odinger operators with oscillating, slowly decaying potentials, revealing non semiclassical phenomena and fewer eigenvalues near zero than classical predictions.

Contribution

It provides new insights into the spectral behavior of Schr"odinger operators with oscillating, decaying potentials, especially near the origin, highlighting deviations from semiclassical expectations.

Findings

Fewer eigenvalues near zero than semiclassical predictions

Identification of non semiclassical phenomena due to irregular decay

Analysis of the asymptotic behavior of the discrete spectrum

Abstract

We consider the Schr\"odinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin, and due to the irregular decay of $\eta W$, we encounter some non semiclassical phenomena. In particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.