A note on Schrödinger equation with linear potential and hitting times

Abstract

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