Spectral Properties of Confining Superexponential Potentials

TL;DR

This paper investigates the spectral characteristics of confining superexponential potentials, revealing complex eigenvalue structures, localization phenomena, and potential applications in cold atom physics.

Contribution

It provides a detailed analysis of eigenvalues and eigenstates of superexponential potentials, including new scaling behaviors and localization effects not previously documented.

Findings

Eigenvalue spacing exhibits scaling behavior and alternates in modified oscillators.

Near-degenerate eigenvalue doublets emerge in symmetric potentials.

Eigenstates are localized in outer wells and form even-odd pairs.

Abstract

We explore the spectral properties and behaviour of confining superexponential potentials. Several prototypes of these highly nonlinear potentials are analyzed in terms of the eigenvalues and eigenstates of the underlying stationary Schr\"odinger equation up to several hundreds of excited states. A generalization of the superexponential self-interacting oscillator shows a scaling behaviour of the spacing of the eigenvalues which turns into an alternating behaviour for the power law modified oscillator. Superexponential potentials with an oscillating power show a very rich spectral structure with varying amplitudes and wave vectors. In the parity symmetric case doublets of near degenerate energy eigenvalues emerge in the spectrum. The corresponding eigenstates are strongly localized in the outer wells of the potential and occur as even-odd pairs which are interspersed into the spectrum of delocalized states. We provide an outlook on future perspectives including the possibility to use these features for applications in e.g. cold atom physics.