# The Schrodinger Equation as a Volterra Problem

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## Date

2011-08-08

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## Abstract

The objective of the thesis is to treat the Schrodinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress,
the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution
of the integral equation. Similarly, the Schrodinger equation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of
the Rubinsteins and Kress to show for the Schrodinger equation similar results to those for the heat equation. The thesis proves that the Schrodinger equation with
a source function does indeed have a unique solution. The Poisson integral formula with the Schrodinger kernel is shown to hold in the Abel summable sense. The Green functions are introduced in order to obtain a representation for any function which satisfies the Schrodinger initial-boundary value problem. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact Green function as n goes to infinity. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces, and the Volterra and General Volterra theorems are proved and used in order to show that the Neumann series for the L^1 kernel, the L^infinity kernel, the Hilbert-Schmidt kernel, the unitary kernel, and the
WKB kernel converge to the exact Green function. In the WKB case, the solution of the Schrodinger equation is given in terms of classical paths; that is, the multiple scattering expansions are used to construct from, the action S, the quantum Green function. Then the interior Dirichlet problem is converted into a Volterra integral
problem, and it is shown that Volterra integral equation with the quantum surface kernel can be solved by the method of successive approximations.

## Description

## Keywords

Schrödinger equation, Volterra integral equations, semiclassical mechanics, WKB approximation