A Fundamental Solution to the Schr\"odinger Equation with Doss Potentials and its Smoothness

We construct a fundamental solution to the Schr"odinger equation for a class of potentials of polynomial type by a complex scaling approach as in [Doss1980]. The solution is given as the generalized expectation of a white noise distribution. Moreover, we obtain an explicit formula as the expectation of a function of Brownian motion. This allows to show its differentiability in the classical sense. The admissible potentials may grow super-quadratically, thus by a result from [Yajima1996] the solution does not belong to the self-adjoint extension of the Hamiltonian.