The spectral determinant for second order elliptic operators on the real line

TL;DR

This paper derives a formula for the spectral determinant of second-order elliptic operators on the real line, enabling explicit calculations for harmonic and anharmonic oscillators with bounded potentials.

Contribution

It provides a new expression for the spectral determinant in terms of Wronskians, applicable to a range of elliptic operators on the real line.

Findings

Explicit formula for spectral determinant using Wronskians

Application to harmonic and anharmonic oscillators

Calculation of determinants with bounded potentials

Abstract

We derive an expression for the spectral determinant of a second-order elliptic differential operator $\mathcal{T}$ defined on the whole real line, in terms of the Wronskians of two particular solutions of the equation $\mathcal{T} u=0$. Examples of application of the resulting formula include the explicit calculation of the determinant of harmonic and anharmonic oscillators with an added bounded potential with compact support.