Convergence of the Neumann series for the Schrodinger equation and general Volterra equations in Banach spaces

TL;DR

This paper establishes the convergence of the Neumann series for solving the Schrödinger equation and general Volterra equations in Banach spaces, extending classical methods with new integral theorems and convergence proofs.

Contribution

It introduces generalized Volterra theorems and proves the convergence of Neumann series for Schrödinger equations in Banach spaces, using an extended Picard iteration approach.

Findings

Neumann series converge in L^p(B) topology for Banach space-valued functions.

Generalized Volterra theorems are proved for arbitrary Banach spaces.

Convergence of the Green function via Neumann series for Hilbert-Schmidt and unitary kernels.

Abstract

The objective of the article is to treat the Schr\"{o}dinger equation in parallel with a standard treatment of the heat equation. In the mathematics literature, 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. The Poisson Integral theorem shows that the Poisson integral formula with the Schrodinger kernel holds in the Abel summable sense. Furthermore, the Source integral theorem provides the solution of the initial value problem for the nonhomogeneous Schrodinger equation. Folland's proof of the Generalized Young's inequality is used as a model for the proof of the L^p lemma. Basically the Generalized Young's theorem is in a more general form where the functions take values in an arbitrary Banach space. The L^1, L^p and the L^\infty lemmas are inductively applied to the proofs of their respective Volterra theorems in order to prove that the Neumman series converge with respect to the topology L^p(I;B}, where I is a finite time interval, B is an arbitrary Banach space, and 1 =< p =< \infty. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact solution as n->\infty. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces. The Volterra theorems are proved and applied in order to show that the Neumann series for the Hilbert-Schmidt kernel and the unitary kernel converge to the exact Green function.