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On the New Form of Representing Cauchy Problem for Schrodinger Equation on the Real Time

Authors: Grishin D.V., Pavlovskiy Ya.Yu., Remizov I.D., Rozhkova E.S., Samsonov D.A. Published: 14.02.2017
Published in issue: #1(70)/2017  
DOI: 10.18698/1812-3368-2017-1-26-42

 
Category: Mathematics | Chapter: Mathematical Physics  
Keywords: Schrodinger equation, Cauchy problem, quasi-Feynman formula, heat equation, Chernoff tangency, multiple integral, operator semigroup

The study examines Cauchy problem for the one-dimensional Schrodinger equation ψ't(f,x) = iHψ(f,x) with the Hamiltonian -H of the form -Hf = 1/2f" + Vf, where potential V is a real-valued differentiable function bounded with its derivative. This equation has been studied from the creation of quantum mechanics, and it still appears to be a good model case for various methods of solving partial differential equations. In this paper we study the problem of representability of the solution of Cauchy problem in the form of the quasi-Feynman formula, and provide a positive answer to this problem. The quasi-Feynman formula constructed in the paper is a new type of expression, similar to the Feynman formula. It includes multiple integrals of an infinitely increasing multiplicity. The quasi-Feynman formulas are easier to prove (compared to the Feynman formulas) but they provide lengthier expression for the solution. The paper may be of interest to the ones who work in the fields of functional analysis and mathematical physics.

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