Le problème de Dirichlet pour l'opérateur de Schrödinger (Q1061920)

The aim of this paper is to prove regularity conditions (up to the boundary) of the Poisson kernel for \(H=-\Delta +V\) in a ball B, V being a potential in the class \(K^{\nu}\) (a very general hypothesis). The main result is that the Poisson kernel for \(-\Delta +V\) is the product of the usual Poisson kernel (that is for \(H_ 0=-\Delta)\) by a function \(K_ V(x,\theta)\) continuous on \(\bar B\times \partial B.\) The method (which uses Brownian motion) can be used to prove similar results about the resolvent of H (with Dirichlet boundary condition): If Re \(\lambda\) \(<\lambda_ 0=\inf sp(H)\), \((H-\lambda)^{-1}(x,y)=K_ V(x,y,\lambda)H_ 0^{-1}(x,y)\) where \(K_ V(x,y,\lambda)\) is a continuous function on \(\bar B\times \bar B\times \{Re \lambda <\lambda_ 0\}.\) Estimates of eigenfunctions at the boundary \((\phi_ n(x)\leq C_ nd(x,\partial B))\) are also obtained.