A trace formula for differential operators of arbitrary order

TL;DR

This paper derives a trace formula for differential operators of any order with rapidly decaying coefficients, expressing the resolvent difference in terms of Wronskians and providing a new representation for the perturbation determinant.

Contribution

It introduces a general trace formula for arbitrary order differential operators with decaying coefficients, linking resolvent differences to Wronskians and perturbation determinants.

Findings

Derived a trace formula for operators of arbitrary order.

Expressed resolvent differences via Wronskians of solutions.

Provided a new representation for the perturbation determinant.

Abstract

An operator $H=H_{0}+V$ where $H_{0}=i^{-N} \partial^{N}$ ($N$ is arbitrary) and $V$ is a differential operator of order $N-1$ with coefficients decaying sufficiently rapidly at infinity is considered in the space $L^2(\Bbb R)$. The goal of the paper is to find an expression for the trace of the difference of the resolvents $(H-z)^{-1}$ and $(H_{0}-z)^{-1}$ in terms of the Wronskian of appropriate solutions to the differential equation $Hu=zu$. This also leads to a representation for the perturbation determinant of the pair $H_{0}, H$.