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

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.

arXiv comment31 pages

References in corpus (6)

• Connes-Chern character for manifolds with boundary and eta cochains
• On the Determinant of One-Dimensional Elliptic Boundary Value Problems
• Regular singular Sturm-Liouville operators and their zeta-determinants
• The Analytic Torsion of the cone over an odd dimensional manifold
• Regularizing infinite sums of zeta-determinants
• Cheeger-Mueller Theorem on manifolds with cusps