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On a New Algorithm for the Computation of Enclosures for the Titchmarsh-Weyl m-Function

B. M. Brown, M. S. P. Eastham, D. K. R. McCormack, M. Plum

TL;DR

The paper develops two new algorithms to compute verified enclosures for the Titchmarsh-Weyl $m$-function of Sturm–Liouville-type problems, addressing cases where prior interval-based methods struggle (notably for $q=-x^\alpha$ with $0<\alpha<2$). Central to the approach is an asymptotic analysis based on repeated diagonalization and Levinson-type error control, combined with an interval ODE solver to propagate validated bounds from large $x$ back to the endpoint. For $a=0$, $p=w=1$, and $q=-x^\alpha$ with $\alpha=1,2$, the method yields enclosures for $m(\lambda)$ that agree with known closed-form expressions. The extension to $0<\alpha<2$ uses an auxiliary parameter $\varepsilon$ and a rigorous transfer theorem (Theorem 5.1) to bound the influence of the origin, enabling validated results even with reduced smoothness. Overall, the work broadens the class of potentials for which guaranteed bounds on the $m$-function can be computed, with demonstrated numerical examples and practical interval-based verification.

Abstract

The paper reports on computation of verified enclosures for the Titchmarsh-Weyl m-function. It examines some cases in which Lohner's AWA algorithm must be suplimented by mathematical analysis.

On a New Algorithm for the Computation of Enclosures for the Titchmarsh-Weyl m-Function

TL;DR

The paper develops two new algorithms to compute verified enclosures for the Titchmarsh-Weyl -function of Sturm–Liouville-type problems, addressing cases where prior interval-based methods struggle (notably for with ). Central to the approach is an asymptotic analysis based on repeated diagonalization and Levinson-type error control, combined with an interval ODE solver to propagate validated bounds from large back to the endpoint. For , , and with , the method yields enclosures for that agree with known closed-form expressions. The extension to uses an auxiliary parameter and a rigorous transfer theorem (Theorem 5.1) to bound the influence of the origin, enabling validated results even with reduced smoothness. Overall, the work broadens the class of potentials for which guaranteed bounds on the -function can be computed, with demonstrated numerical examples and practical interval-based verification.

Abstract

The paper reports on computation of verified enclosures for the Titchmarsh-Weyl m-function. It examines some cases in which Lohner's AWA algorithm must be suplimented by mathematical analysis.

Paper Structure

This paper contains 9 sections, 1 theorem, 55 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Titchmarsh-Weyl limit-point, limit-circle classification
  3. Overview of the components of the algorithm
  4. Asymptotic theory
  5. Approximate diagonalization
  6. Interval ODE solver
  7. Results for $q=-x^\alpha,\;\;\alpha =\;1,\; 2$
  8. Results for $q=-x^\alpha$,$0 < \alpha < 2$, $a=0,p=w=1$
  9. Numerical examples

Key Result

Theorem 5.1

Let $c(x)=q(x)-\lambda$ and, for some $\epsilon >0$, let $f,g \in C_1[0,\epsilon]$ satisfy $f(\epsilon)=y(\epsilon)$, $g(\epsilon)=y^{'}(\epsilon)$. In addition let Then

Theorems & Definitions (1)

  • Theorem 5.1