Spectral properties of the Cauchy-Poisson problem and the local asymptotic behavior of waves (Q1093092)

The author considers the Cauchy problem for the equation \[ - u_{tt}(x,\nu,t)+[H(\nu)u(\cdot,\nu,t)](x)=0, \] where H(\(\nu)\) is a non-negative operator in \(L^ 2({\mathbb{R}}_ x)\) with an absolutely continuous spectrum in \([\lambda_ 0(\nu),\infty)\). The function u(x,\(\nu\),t) is a Fourier transform of the function \(\phi\) (x,0,z,t) with respect to z and \(\phi\) is a potential of a velocity field. The author investigates the analytical properties of the resolvent of the operator H(\(\nu)\). A meromorphic continuation of the resolvent of H(\(\nu)\) in the upper halfplane Im \(\mu\geq 0\) is obtained and the asymptotic behaviour of the function \(\phi\) (x,0,vt,t) as \(t\to \infty\) is investigated.