POINTWISE PROPERTIES OF EIGENFUNCTIONS AND HEAT KERNELS OF DIRICHLET-SCHRODINGER OPERATORS

Let D subset of R-d be an open domain, H-o = Sigma(i,j=1)(d) partial d erivative(i)(a(i,j)(x)delta(j)) a second order elliptic operator with continuous coefficients, and let H = H-o + q (q : D bar right arrow [O , infinity)) be a Schrodinger operator associated with H-o, acting on L-2(D), with Dirichlet boundary conditions. We provide in this paper b oth L-2 and pointwise bounds for the eigenfunctions of H, in term of t he Agmon's metric of q and of the quasi-hyperbolic geometry of D. At l east when H-o = -Delta, we show that the pointwise bounds obtained for the ground state eigenfunction of H are qualitatively sharp either wh en q diverges sufficiently fast at the boundary, or in planar regular domains. We also give applications to the intrinsic ultracontractivity of H. Finally, we prove a result concerning the pointwise decay at th e boundary of the heat kernel of H in a-regular domains.