Superharmonic double-well systems with zero-energy ground states: Relevance for diffusive relaxation scenarios

TL;DR

This paper investigates the spectral properties of a class of superharmonic double-well Schrödinger systems related to diffusive relaxation, developing a computational method to approximate their spectra, especially for large potential exponents.

Contribution

It introduces a computer-assisted approach to approximate the low-energy spectrum of superharmonic double-well systems with zero-energy ground states, addressing a gap in spectral data for these quasi-exactly solvable models.

Findings

Developed a numerical procedure for spectral approximation of $\, ext{H}$.

Analyzed spectral closeness between $ ext{H}$ and Neumann Laplacian for large $m$.

Extended spectral analysis to potentials with $m$ up to 104.

Abstract

Relaxation properties (specifically time-rates) of the Smoluchowski diffusion process on a line, in a confining potential $ U(x) \sim x^m$, $m=2n \geq 2$, can be spectrally quantified by means of the affiliated Schr\"{o}dinger semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - \Delta + {\cal{V}}(x)$ involves a potential function ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$, which for $m>2$ has a conspicuous higher degree (superharmonic) double-well form. For each value of $m>2$, $ \hat{H}$ has the zero-energy ground state eigenfunction $\rho _*^{1/2}(x)$, where $\rho _*(x) \sim \exp -[U(x)]$ stands for the Boltzmann equilibrium pdf of the diffusion process. A peculiarity of $\hat{H}$ is that it refers to a family of quasi-exactly solvable Schr\"{o}dinger-type systems, whose spectral data are either residual or analytically unavailable. As well, no numerically assisted procedures have been developed to this end. Except for the ground state zero eigenvalue and incidental trial-error outcomes, lowest positive energy levels (and energy gaps) of $\hat{H}$ are unknown. To overcome this obstacle, we develop a computer-assisted procedure to recover an approximate spectral solution of $\hat{H}$ for $m>2$. This task is accomplished for the relaxation-relevant low part of the spectrum. By admitting larger values of $m$ (up to $m=104$), we examine the spectral "closeness" of $\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta _{\cal{N}}$ in the interval $[-1,1]$, known to generate the Brownian motion with two-sided reflection.