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) = R 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 = − 1 2 d dx a d dx + Vb,a, where Vb,a = 1 2 ( 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 σ(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.