Asymptotics of the spectral data of perturbed Stark operators in the half-line with mixed boundary conditions

TL;DR

This work analyzes spectral data for perturbed Stark operators on the half-line with mixed boundary conditions. It develops a robust perturbative framework around the unperturbed Airy-based problem, establishing real-analytic dependence of eigenvalues and norming constants on the perturbation q in \mathfrak{A}_r (r>1). The authors derive sharp, uniform asymptotics for the spectral data in terms of Airy-function integrals of q and the unperturbed Ai' zeros, revealing how the Stark term and boundary condition influence the spectrum. The results advance inverse spectral theory for Stark-type operators by linking spectral data to the perturbation in a quantitatively precise way, with explicit remainder estimates governed by the index r and the auxiliary function \omega_r(n).

Abstract

We obtain sharp asymptotic formulas for the eigenvalues and norming constants of Sturm-Liouville operators associated with the differential expression \[ -\frac{d^2}{dx^2} + x + q(x), \quad x\in [0,\infty), \] together with the boundary condition $\varphi'(0) - b\varphi(0) =0$, $b\in\mathbb{R}$, where \[ q\in \left\{ p\in L^2_{\mathbb{R}}(\mathbb{R}_+,(1+x)^r dx) : p'\in L^2_{\mathbb{R}}(\mathbb{R}_+,(1+x)^r dx)\right\} \] with $r>1$.

Theorems & Definitions (44)

• Proposition
• Theorem
• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5