Potential Analysis 9: 65–82, 1998. 65 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. On the Schr ¨ odinger Equation with Potentials which are Laplace Transforms of Measures S. ALBEVERIO 1 , Z. BRZE ´ ZNIAK 1 and Z. HABA 2 1 Fakult¨ at f ¨ ur Mathematik, Ruhr-Universit¨ at Bochum, Germany 2 Institute for Theoretical Physics, University of Wroclaw, Poland (Received: 12 January 1996; accepted: 16 September 1996) Abstract. We construct a pointwise solution for the time dependent Schr¨ odinger equation on with potentials and initial conditions which can grow exponentially at infinity and belong to the class of smooth Laplace transforms of complex measures on . The methods used are both analytic and probabilistic and the result can be looked upon as an extension of rigorously defined Feynman path integrals to the case of potentials which can strongly grow at infinity. An appendix with the calculation of some Wiener integrals is also presented. Mathematics Subject Classifications (1991): 58D30, 35Q40, 81Q20, 46N50, 58C27, 81Q05. Key words: Schr¨ odinger equation, Feynman–Kac formula, Laplace transform. 1. Introduction The study of solutions of the time dependent Schr¨ odinger equation is of basic importance in quantum mechanics. There are analytic results on existence and uniqueness under several types of assumptions on the potentials, most results being discussed in a 2 -setting, see e.g. [14, 22, 28]. The 2 -setting is of importance for the usual physical interpretation of the solutions, however solutions outside 2 have also been discussed in several contexts, e.g. in the space of quasi-periodic functions (see e.g. [15]), -spaces (see e.g. [8,17]), in connection with certain explicitly solvable models (see e.g. [8, 16]). The 2 -analytic methods work especially well in the case where the potential is such that the Hamiltonian is lower bounded and essentially self-adjoint on some suitable minimal domain (e.g. smooth functions with compact support). In the case of strong local singularities or potentials which are unbounded at infinity (i.e. with a singularity at infinity), leading to an Hamiltonian which is unbounded from below, other methods have to be involved. Here mathematical functional integration methods of both basic variants, probabilistic integrals (of Wiener type [26, 29], associated with Dirichlet processes [2, 10] or, see e.g. [1, 24] SFB 237 (Essen–Bochum–Dusseldorf); BiBoS; CERFIM (Locarno). Supported by DFG, current address: Department of Pure Mathematics, The University of Hull, Hull HU6 7RX, UK.