谷島 賢二

We report our recent results on the existence and uniqueness of unitary propagators for [Formula: see text]-particle Schrödinger equations which may be applied to most interesting problems in physics.

We study the threshold behavior of two-dimensional Schrödinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with [Formula: see text] or [Formula: see text]-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centers. When the operator is of regular type, we prove that the wave operators are bounded in [Formula: see text] for all [Formula: see text]. With a single center, we always are in the regular type case.

We show that the fundamental solution of the initial value problem for the time dependent SchrÖdinger equation is bounded and continuous for a class of non-smooth potentials. The class is large enough to accomodate Coulomb potentials if the spatial dimension is three.