Quantum square well with logarithmic central spike

TL;DR

This paper investigates a linear Schrödinger equation with a logarithmic spike at the origin, providing perturbative and non-perturbative insights into its spectrum and singularity structure, inspired by nonlinear Schrödinger models.

Contribution

It introduces a novel linear model with a logarithmic central spike, analyzing its spectral properties and singularity behavior using perturbation theory and variable transformation.

Findings

Weak-coupling regime well-described by Rayleigh-Schrödinger perturbation theory

First-order spectrum obtained in closed form

Non-perturbative analysis reveals the nature of the singularity at zero

Abstract

Linear square-well Schr\"{o}dinger equation endowed with a singular logarithmic spike in the origin is studied. The study is methodical, motivated by the problem of non-gausson states $\psi_n(x)$, $n \neq 0$ generated by nonlinear Schr\"{o}dinger equations. Once the state-dependent self-interaction term is chosen logarithmic, $\sim -g\,\ln[\psi^*_n(x)\psi_n(x)]$, the nonlinear model develops the puzzling logarithmic (i.e., weakly singular) repulsive barriers near the nodal zeros of $\psi_n(x)$ at $n \neq 0$. In our linearized approach the weak-coupling regime is shown reliably described by the routine Rayleigh-Schr\"{o}dinger perturbation theory. It even provides the first-order picture of the spectrum in closed-form. Beyond the weak-coupling regime an amendment of the unperturbed Hamiltonian is recommended. Finally, an analytic insight into the nature of the singularity at $x=0$ is obtained, in a non-perturbative setting, after the change of variables $x=\exp y$.