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

TL;DR

This paper constructs a fundamental solution for the Schrödinger equation with polynomial-type potentials using complex scaling, providing explicit formulas and demonstrating classical differentiability, even for super-quadratic growth potentials.

Contribution

It introduces a novel complex scaling method to explicitly construct and analyze the smoothness of solutions for Schrödinger equations with broad classes of polynomial potentials.

Findings

Explicit fundamental solution expressed as a white noise expectation.

Demonstrates classical differentiability of the solution.

Applicable to super-quadratic growth potentials, extending previous results.

Abstract

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.