A Fundamental Solution to the Schrödinger Equation with Doss Potentials and its Smoothness

2015/03/17 by Martin Grothaus, Grothaus, Martin, Felix Riemann +1
Mathematics · Physics and Astronomy · #46G12 #46N50 #60H40 #81S40 #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Statistical Mechanics and Entropy #math-ph #math.MP #msc:46G12 #msc:46N50 #msc:60H40 #msc:81S40

paper · pdf · doi:10.48550/arxiv.1503.05058

32 pages

arxiv created 2015/03/17 · openalex publication_date 2015/03/17 · arxiv updated 2015/03/18 · openalex created_date 2022/08/18 · openalex updated_date 2026/07/28

Abstract

We construct a fundamental solution to the Schrödinger 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.

Related

• Feynman integrals as Hida distributions: the case of non-perturbative potentials
• Short time asymptotics of the fundamental solutions for Schrödinger equations with non-smooth potentials
• Non-smoothness of the fundamental solutions for Schrödinger equations with super-quadratic and spherically symmetric potential
• The Schroedinger Equation with Potential in Random Motion
• General Solution of the Schrödinger Equation with Potential Field Quantization