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Complex oscillations of non-definite Sturm-Liouville problems, II

Angelo B. Mingarelli

TL;DR

The paper addresses non-definite Sturm-Liouville problems with complex ghost eigenfunctions $y=\varphi+i\psi$, correcting Richardson's separation result and extending it to weights $r(x)$ that vanish on subintervals away from the endpoints. It introduces Theorem th3, which replaces the original separation by allowing an interval of zeros and analyzes the sign of $G(x)=∫_a^x r|y|^2 dt$ to obtain interlacing of zeros under weight-sign-change patterns. Through several examples (including piecewise-constant weights and Airy-function constructions) the authors demonstrate sharpness, show endpoint-inclusion failures, and discuss open questions. The work also develops Sturm-theoretic consequences for non-real eigenfunctions, establishing finiteness of non-real eigenvalue pairs and clarifying when interior zeros must occur, with connections to prior results and broader open problems.

Abstract

We correct and update a result of R.G.D. Richardson [13] dealing with the separation of zeros of the real and imaginary parts of non-real eigenfunctions of non-definite Sturm-Liouville eigenvalue problems. We then extend it to the case where the weight function is allowed to be identically zero on a subinterval that excludes the end-points and study the behavior of the zeros of the real and imaginary parts when the end-points are included. Examples are given illustrating the sharpness of the results along with open questions.

Complex oscillations of non-definite Sturm-Liouville problems, II

TL;DR

The paper addresses non-definite Sturm-Liouville problems with complex ghost eigenfunctions , correcting Richardson's separation result and extending it to weights that vanish on subintervals away from the endpoints. It introduces Theorem th3, which replaces the original separation by allowing an interval of zeros and analyzes the sign of to obtain interlacing of zeros under weight-sign-change patterns. Through several examples (including piecewise-constant weights and Airy-function constructions) the authors demonstrate sharpness, show endpoint-inclusion failures, and discuss open questions. The work also develops Sturm-theoretic consequences for non-real eigenfunctions, establishing finiteness of non-real eigenvalue pairs and clarifying when interior zeros must occur, with connections to prior results and broader open problems.

Abstract

We correct and update a result of R.G.D. Richardson [13] dealing with the separation of zeros of the real and imaginary parts of non-real eigenfunctions of non-definite Sturm-Liouville eigenvalue problems. We then extend it to the case where the weight function is allowed to be identically zero on a subinterval that excludes the end-points and study the behavior of the zeros of the real and imaginary parts when the end-points are included. Examples are given illustrating the sharpness of the results along with open questions.

Paper Structure

This paper contains 4 sections, 16 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Sturm theory for non-real eigenfunctions

Key Result

Theorem 1

[ Rich18, Theorem X] Let $r$ be continuous in $[a, b]$. If $r(x)$ changes its sign precisely once in $(a,b)$ then the roots of the real and imaginary parts of any complex ghost separate one another.

Figures (4)

  • Figure 3.1: Here, $\varphi$ is the left-most curve of the graph on the left, $\psi$ being the other. The graph on the right is that of $G(x)$.
  • Figure 3.2: Here, $\varphi$ is the upper curve on the left.
  • Figure 3.3: Here, the left-most curve is $\varphi$, the real part.
  • Figure 3.4: Here, $\varphi$ is the left-most curve of the graph on the left, $\psi$ being the other. Note the positivity of the real part in $(-1,1)$. Again, $G(x)<0$, here.

Theorems & Definitions (38)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • ...and 28 more