Normally, the time evolution in quantum mechanics is given by a unitary one-parameter group. In this talk about an ongoing project, I am pursuing a natural generalization in which a particle may escape to infinity in finite time. A prototype is a classical particle in one dimension with an x3-potential. This confirms the intuitive connection between classical incompleteness and selfadjointness. From the classical analogy it is natural to modify the dynamics to a contraction semigroup. This can either be obtained by the dissipative branch of von Neumann’s extension theory, or by a Trotter limit in the weak operator topology, by which the sum of selfadjoint operators may sometimes be well-defined, but fail to be selfadjoint. For the cubic potential, an approach based on Nelson’s regularization of path integrals yields a specific contraction semigroup.
