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

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