Zeta-determinants of Sturm-Liouville operators with quadratic potentials at infinity

TL;DR

This paper studies zeta-determinants of Sturm-Liouville operators with quadratic growth potentials on half-lines, providing formulas in terms of Wronskians, with applications to hyperbolic geometry and spectral analysis.

Contribution

It establishes existence and explicit formulas for zeta-determinants of such operators, linking spectral properties to boundary conditions and fundamental solutions.

Findings

Derived formulas for zeta-determinants in terms of Wronskians.

Connected spectral analysis of these operators to hyperbolic geometry.

Extended understanding of analytic torsion in manifolds with cusps.

Abstract

We consider Sturm-Liouville operators on a half line $[a,\infty), a>0$, with potentials that are growing at most quadratically at infinity. Such operators arise naturally in the analysis of hyperbolic manifolds, or more generally manifolds with cusps. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. Despite being the natural objects in the context of hyperbolic geometry, spectral geometry of such operators has only recently been studied in the context of analytic torsion.