# Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials

## TL;DR

This paper explores the spectral properties of diffusion processes in steep and bistable potential wells, analyzing their convergence, spectral closeness to Laplacians, and the spectral behavior of associated Schrödinger operators with complex potentials.

## Contribution

It provides a detailed spectral analysis of diffusion generators transformed into Schrödinger operators, revealing conditions for spectral closeness and the absence of negative eigenvalues in bistable potentials.

## Key findings

- Spectral closeness of Schrödinger operators to Laplacians for large potential exponents.
- Conditions under which the operators become spectrally close to Dirichlet or Neumann Laplacians.
- Proof of the absence of negative eigenvalues in certain bistable potential Schrödinger operators.

## Abstract

We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials $U(x)\sim x^m$, $m=2n \geq 2$. This is paralleled by a transformation of each $m$-th diffusion generator $L = D\Delta + b(x)\nabla $, and likewise the related Fokker-Planck operator $L^*= D\Delta - \nabla [b(x)\, \cdot]$, into the affiliated Schr\"{o}dinger one $\hat{H}= - D\Delta + {\cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $\exp(tL)$, $\exp(tL_*)$ and $\exp(-t\hat{H})$, with $t \geq 0$. The Feynman-Kac integral kernel of $\exp(-t\hat{H})$ is the major building block of the relaxation process transition probability density, from which $L$ and $L^*$ actually follow. The spectral "closeness" of the pertinent $\hat{H}$ and the Neumann Laplacian $-\Delta_{\cal{N}}$ in the interval is analyzed for $m$ even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the $m$-family of operators $\hat{H}$ with a priori chosen ${\cal{V}}(x) \sim x^m$, when $ \hat{H}$ becomes spectrally "close" to the Dirichlet Laplacian $-\Delta_{\cal{D}}$ for large $m$. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for $\hat{H}$ with a bistable-looking potential ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}, a, b, >0, m>2$ has been addressed.