Schr\"odinger equation with linear potential and hitting times

TL;DR

This paper explores solutions to Schr"odinger's equation with a linear potential related to hitting times of Brownian motion, addressing boundary condition challenges in previous work using an alternative approach.

Contribution

It introduces a method to find solutions satisfying both the PDE and boundary conditions at t=0, improving upon prior limitations.

Findings

Successfully constructs solutions satisfying PDE and boundary conditions at t=0

Extends understanding of Schr"odinger equations with linear potentials

Provides a new approach for boundary value problems in stochastic PDEs

Abstract

In Hern\'andez-del-Valle (2010) the author studies the connection between Schr\"odinger's equation and first hitting densities of Brownian motion. Although the author is able to find solutions of a Schr\"odinger type pde he fails---except in some special cases---to construct a solution which satisfies the boundary on the space variable at $x=0$. In this paper we use an approach used in Bluman and Shtelen (1996) to find solutions which satisfy the pde and boundary condition when $t=0$.