Explicit and almost explicit spectral calculations for diffusion operators

The diffusion operatorHD=−12ddxaddx−bddx=−12exp(−2B)ddxaexp(2B)ddx, where B(x)=∫0xba(y)dy, defined either on R+=(0,∞)R+=(0,∞) with the Dirichlet boundary condition at x=0x=0, or on RR, 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=−12ddxaddx+Vb,a, where Vb,a=12(b2a+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)infσ(HD) scales when a=νa0a=νa0 and b=γb0b=γb0, where ν and γ   are parameters, and a0a0 and b0b0 are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators.