Low energy resolvent bounds for elliptic operators: an application to the study of waves in stratified media and fiber optics

Abstract

For the positive self-addjoint operators on with n ≥ 3 it is shown that as Re(z) → 0⁺, by imposing several conditions on g_ij and q. In the special case g_ij = c²δ_ij these conditions reduce to with the nontrapping condition and a positivity condition , for some κ, C, ρ > 0. Results are applied to the stratified wave equation , where with n = κ + m, and . In all cases the condition (c-y.▽_yc)≥kc leads to a local-energy decay estimate for ψ(z,t).