Boundedness and continuity of the fundamental solution of the time dependent Schrödinger equation with singular potentials (Q1283240)

Let \(E(t,x,y)\) be the fundamental solution of the time-dependent Schrödinger equation \[ i{\partial u \over \partial t} = -{1\over 2}\Delta u + V(x)u, \quad (t,x) \in {\mathbb R}^1 \times {\mathbb R}^m; \qquad u(0,x) = u_0(x), \quad x \in {\mathbb R}^m. \] For a class of non-smooth potentials \(V\) it is proved that the fundamental solution is bounded and continuous with respect to \((t,x,y)\) for \(t \neq 0\). This class of potentials is large enough to accommodate Coulomb potentials \(V(x) = \sum_{j=1}^N {Z_j \over | x-R_j| }\) in dimension three.