On traces and modified Fredholm determinants for half-line Schr\"odinger operators with purely discrete spectra

TL;DR

This paper explores the relationship between traces and modified Fredholm determinants for a class of half-line Schrödinger operators with potentials diverging at infinity, providing new insights into their spectral properties.

Contribution

It applies a fundamental trace-Fredholm determinant identity to Schrödinger operators with rapidly diverging potentials, extending spectral analysis techniques to this class.

Findings

Established a trace-Fredholm determinant relation for these operators

Analyzed spectral properties under various boundary conditions

Extended existing methods to potentials diverging faster than polynomial rates

Abstract

After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schr\"odinger operators $(- d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$.