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Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions

Vladislav V. Kravchenko

Abstract

A new representation for solutions of the one-dimensional Schrödinger equation -u"+q(x)u=w^2u is obtained in the form of a series possessing the following attractive feature. The truncation error is w-independent for all real w. For the coefficients of the series simple recurrent integration formulas are obtained which make the new representation applicable for computation.

Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions

Abstract

A new representation for solutions of the one-dimensional Schrödinger equation -u"+q(x)u=w^2u is obtained in the form of a series possessing the following attractive feature. The truncation error is w-independent for all real w. For the coefficients of the series simple recurrent integration formulas are obtained which make the new representation applicable for computation.

Paper Structure

This paper contains 5 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Construction of a transmutation
  3. A representation for solutions of (\ref{['Schr']})
  4. A recurrent procedure for computing the coefficients $a_{n}$
  5. Numerical illustrations

Key Result

Proposition 4

Theorems & Definitions (10)

  • Definition 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Proposition 9
  • Example 10