A new approach to inverse Sturm-Liouville problems based on point interaction II. The singular case

TL;DR

This work extends inverse Sturm–Liouville theory to singular problems on the half-line by embedding $\delta$-point interactions to generate a family of perturbed problems whose first eigenvalue function $\lambda(t,r;q)$ encodes the unknown potential $q$. Under two regimes—$({\bf H_1})$ with $q\ge0$ (limit-point at $+\infty$) and $({\bf H_2})$ with limit-circle and non-oscillatory behavior—the first eigenvalue exists as a principal eigenvalue for perturbations, and $\lambda(t,r;q)$ is continuous and differentiable in $(t,r)$ with explicit derivative relations. The main result yields a unique reconstruction of $q$ via $q(x)=\frac{\varphi''_0(x)}{\varphi_0(x)}+\lambda_1$, where $\varphi_0(x)=\sqrt{-\frac{\partial \lambda(x,0)}{\partial r}}$ and $\lambda_1=\lambda(t,0)$, linking the potential to the sensitivity of the first eigenvalue to point perturbations. The framework leverages Weyl theory, perturbation analysis, and norm-resolvent convergence to provide a direct, explicit recovery on the non-compact domain and delineates conditions for the method’s applicability.

Abstract

In this paper, further to the point interaction method for inverse Sturm-Liouville problems on finite intervals firstly proposed in our previous work, we will continue to generalize this method to the inverse eigenvalue problems for singular Sturm-Liouville problems on the half real axis.

Theorems & Definitions (15)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Lemma 3.4
• Theorem 3.5
• Lemma 3.6