Inverse spectral problem for analytic domains I: Balian-Bloch trace formula

This is the first in a series of papers [Z3, Z4] on inverse spectral/resonance problems for analytic plane domains ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BEI, BB2], It is an asymptotic formula for the trace Tr l ω R_ρ (k+i τ log k) of the regularized resolvent of the Dirichlet or Neumann Laplacian of ω as k → ∞ with τ > 0. When the support of p̂ contains the length L_Y of precisely one periodic reflecting ray y, then the asymptotic expansion of T r 1 ω R_ρ (k + i τ log k) is essentially the same as the wave trace expansion at y. The raison d'ètre for this approach is that it leads to relatively simple explicit formulae for wave invariants. For example, we give the first formulae for wave invariants of bouncing ball orbits of plane domains (the details will appear in [Z3]). Although we only present details in dimension 2, the methods and results extend with few modifications to all dimensions.