Explicit and almost explicit spectral calculations for diffusion operators

The diffusion operator HD =− 1 2 d dx a d dx − b d dx =− 1 2 exp(−2B) d dx a exp(2B) d dx , where B(x) = x 0 b a (y) dy, defined either on R+ = (0,∞) 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 HS =−12d dx a d dx +Vb,a, where Vb,a = 12 ( b2 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 infσ(HD) scales when a = νa0 and b = γb0, where ν and γ are parameters, and a0 and b0 are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators. © 2008 Elsevier Inc. All rights reserved.