The spectral asymptotics of elliptic operators of Schr�dinger type on a hyperbolic space

Abstract

For self-adjoint second-order elliptic differential operators that satisfy the non-trapping condition on the n-dimensional hyperbolic spaceHnand coincide with the operator\( - \Delta - \left( {\frac{{n - 1}}{2}} \right)^2 \)in a neighborhood of infinity, where Δis the Laplace-Beltrami operator onHn,we obtain the complete asymptotic expansion of the spectral function as λ → +∞.For self-adjoint operators of the form (−Δ)\(\left( { - \Delta } \right)^{m/2} \)+Qm−r,where Qm−r is a pseudodifferential operator of order m−r that is automorphic with respect to a discrete group of isometries of the spaceHnwhose fundamental domain has finite volume, we introduce the spectral distribution function N(λ),which is the analog of the integrated state density, and we find its asymptotics up to order O(λ(n−r)/m)as λ → +∞.Bibliography: 49 titles.