Spectral inclusion and spectral exactness for singular non-selfadjoint Sturm-Liouville problems

TL;DR

This work extends spectral inclusion and spectral exactness theory to singular non-selfadjoint Sturm–Liouville problems by analyzing interval-truncation regularization through the Sims classification. It develops a framework based on the $m$-function and its convergence to determine when truncated problems approximate the original spectrum, and provides a simple, practical test to detect spectral inexactness in Sims Case I. For Sims Cases II and III, norm-resolvent convergence yields spectral inclusion and exactness, while for Case I a targeted test detects spurious eigenvalues arising from regularization. The numerical examples illustrate these concepts and highlight how spurious modes can be identified, with implications for reliable resonance and stability computations in non-selfadjoint settings.

Abstract

We consider the effect of regularization by interval truncation on the spectrum of a singular non-selfadjoint Sturm-Liouville operator. We present results on spectral inclusion and spectral exactness for the cases where the singularity is in Sims Case II or Sims Case III. For Sims Case I we present a test for spectral inexactness, which can be used to detect when the interval truncation process is generating spurious eigenvalues. Numerical results illustrate the effectiveness of this test.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Definition 4.1
• Definition 4.2