Resolvents and Martin boundaries of product spaces (Q1865764)

The authors use geometric scattering theory to study the Laplacian on the product \(X\) of two conformally compact manifolds \((\overline M_1,g_1)\) and \((\overline M_2,g_2)\) which are asymptotically hyperbolic; such manifolds are modeled on the product of hyperbolic spaces near the corners \(\partial\overline M_1\times\partial\overline M_2\). They describe the asymptotic behavior of the resolvent on the Schwarz class by logarithmically blowing up each factor and then performing a standard blow-up of the corner of the product. They examine the asymptotic behavior of the resolvent kernel and determine the Martin boundary of \(X\); this experiences a substantial collapse if there are \(L^2\) eigenvalues.