A note on Schr\"odinger equation with linear potential and hitting times

TL;DR

This paper derives a solution to a Schrödinger-type backward equation with a linear potential, relevant for hitting-time problems, using a method that is independent of a specific parameter, and proposes a way to generate additional solutions.

Contribution

It introduces a novel solution to the Schrödinger-type backward equation with linear potential, applicable to hitting-time problems, and suggests a procedure for generating more solutions.

Findings

Derived a solution satisfying boundary conditions for hitting-time problems.

The solution is independent of the parameter λ.

Proposed a method to generate additional solutions.

Abstract

In this note we derive a solution to the Schr\"odinger-type backward equation which satisfies a necessary boundary condition used in hitting-time problems [as described in Hern\'andez-del-Valle (2010a)]. We do so by using an idea introduced by Bluman and Shtelen (1996) which is worked out in Hern\'andez-del-Valle (2010b). This example is interesting since it is independent of the parameter $\lambda$, namely: \kappa(s,x)=\frac{x}{\sqrt{2\pi s}}\exp{-\frac{(x+\int_0^sf'(u))^2}{2s}}. and suggest a procedure to generating more vanishing solutions and $x=0$.