A note on Schrödinger equation with linear potential and hitting times

2010/09/03 by Gerardo Hernández‐del‐Valle, Gerardo Hernández-del-Valle, Hernández-del-Valle, Gerardo
Mathematics · Physics and Astronomy · #45D05 #45G10 #45G15 (Secondary) #45K05 #45Q05 #60J60 #60J65 (Primary) #Advanced Mathematical Physics Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.PR #msc:45D05 #msc:45G10 #msc:45G15 #msc:45K05 #msc:45Q05 #msc:60J60 #msc:60J65

paper · pdf · doi:10.48550/arxiv.1009.0730

arxiv created 2010/09/03 · openalex publication_date 2010/09/03 · arxiv updated 2010/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 λ, namely: κ(s,x)=(x)/(√(2πs))exp-((x+∫₀^sf'(u))²)/(2s). and suggest a procedure to generating more vanishing solutions and x=0.

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