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Aizenman and Simon show how to solve the Dirichlet problem on flat spaces for a very wide class of potentials here using the intuitive Feynman-Kac representation $(M_V f)(x) = E_x[ \exp(-\int_0^T V(b(s))ds) f(b(T))]$ where $T$ is the first exit time and $f(x)$ is a function on the boundary.  There is little difficulty in getting the same representation for a Riemannian manifold where the labor of showing that the Wiener measure exists and behaves correctly is the subject of many people’s efforts.  One can consider the Schroedinger eigenfunction problem by considering $V' = V - \mu$.  In general, I am interested not in the widest possible set of potentials but rather in understanding whether the right sort of Schroedinger operators in a 4-sphere could have better behavior than the standard Coulomb potential used in flat 3-space.