A trace formula for differential operators of arbitrary order

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 $L^2(\mathbb{R})$. 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 $H_0,H$.

26 pages