Direct proof of energy conservation for automorphic wave equation

The resonances in the problem of scattering on the fundamental domain of the modular group are related to the zeros of Riemann's $\zeta$ function on the critical line [1]. Therefore, the rate of decrease of the energy in the solution given by the Eisenstein series on the translationally invariant subspace is determined by the position of the zeros of the $\zeta$ function. Decrease of the energy can be expected only if there is mutual compensation of the terms of the series [2]. The question of corresponding compensations in the simpler situation in the complete space is therefore of interest.