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

TL;DR

This paper derives precise asymptotic formulas for eigenvalues and norming constants of perturbed Stark operators on the half-line with mixed boundary conditions, considering potentials with specific weighted Sobolev regularity.

Contribution

It provides sharp asymptotic formulas for spectral data of Sturm-Liouville operators with linear potential and weighted Sobolev potentials, extending spectral theory results.

Findings

Sharp asymptotic formulas for eigenvalues

Asymptotics for norming constants

Extension to potentials with weighted Sobolev regularity

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 $r>1$.