Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

Abstract

The diffusion operator $$H_D=-\frac{1}{2}\frac{d}{dx}a\frac{d}{dx}-b\frac{d}{dx}=-\frac{1}{2}\exp (-2B)\frac{d}{dx}a\exp (2B)\frac{d}{dx},$$ where $B(x)={\int }_0^x\frac{b}{a}(y)dy$, 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 Schrödinger-type operator $H_S=-\frac{1}{2}\frac{d}{dx}a\frac{d}{dx}+V_{b,a}$, where $V_{b,a}=\frac{1}{2}(\frac{b^2}{a}+b^{\prime })$. 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.