Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

TL;DR

This paper derives explicit spectral criteria and formulas for diffusion operators, connecting them to Schrödinger operators, and explores how their spectra scale with parameters, with applications to multi-dimensional cases.

Contribution

It provides explicit spectral bounds and criteria for diffusion operators, including their relation to Schrödinger operators, and analyzes parameter scaling effects.

Findings

Explicit criteria for compact resolvent existence.

Formulas for infimum of spectrum and essential spectrum.

Spectral scaling behavior with parameters.

Abstract

The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The operator is unitarily equivalent to the Schr\"odinger-type operator $H_S=-\frac12\frac d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$. We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how $\inf\sigma(H_D)$ scales when $a=\nu a_0$ and $b=\gamma b_0$, where $\nu$ and $\gamma$ are parameters, and $a_0$ and $b_0$ are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators.