On a conjecture of Bennewitz, and the behaviour of the Titchmarsh-Weyl matrix near a pole

TL;DR

The paper addresses the existence of HELP inequalities for higher-order selfadjoint differential expressions by analyzing poles of the Titchmarsh–Weyl $M$-matrices. It proves Bennewitz's conjecture, showing that if $\operatorname{rank}(\operatorname{Res}(M_D,0))+\operatorname{rank}(\operatorname{Res}(M_N,0))=n$, then a valid inequality exists, with a proof grounded in Laurent expansions and Nevanlinna properties. A novel numerical framework is then developed to compute $M_N$ and $M_D$ near a pole by transforming to a safe variable $\Psi=(\alpha I + M^{-1})^{-1}$, solving an initial-value problem for $\Gamma=\Psi$, and extracting the Laurent coefficients of $M$ from the Taylor data of $\Psi$. The authors apply the method to several fourth-order equations, obtaining residue data that typically exhibits low rank and corroborates the conjecture’s predictions about the existence of HELP inequalities for these problems. Collectively, the work provides both theoretical assurance and practical computational tools for assessing HELP-type inequalities in higher-order settings.

Abstract

For any real limit-$n$ $2n$th-order selfadjoint linear differential expression on $[0,\infty)$, Titchmarsh- Weyl matrices $M(λ)$ can be defined. Two matrices of particu lar interest are the matrices $M_D(λ)$ and $M_N(λ)$ assoc iated respectively with Dirichlet and Neumann boundary conditions at $x=0$. These satisfy $M_D(λ) = -M_{N}(λ)^{-1}$. It is known that when these matrices have poles (which can only lie on the real axis) the existence of valid HELP inequalities depends on their behaviour in the neighbourhood of these poles. We prove a conjecture of Bennewitz and use it, together with a new algorithm for computing the Laurent expansion of a Titchmarsh-Weyl matrix in the neighbourhood of a pole, to investigate the existence of HELP inequalities for a number of differential equations which have so far proved awkward to analyse

Theorems & Definitions (7)

• Conjecture 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1