A trace formula for differential operators of arbitrary order

Jörgen Östensson (1) , Dimitri R Yafaev (2)

1 Department of Mathematics [Uppsala]
2 IRMAR - Institut de Recherche Mathématique de Rennes

An operator H = H0 +V where H0 = i (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 H2(R). The goal of the paper is to find an expression for the trace of the difference of the resolvents (H) -1 and (H0 - 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 H0H.

Mots clés: One-dimensional differential operators; arbitrary order; resolvents; perturbation determinant; trace formula

Domaines: Equations aux dérivées partielles [math.AP]

Jörgen Östensson, Dimitri R Yafaev. A trace formula for differential operators of arbitrary order. Harry Dym, Marinus A. Kaashoek, Peter Lancaster, Heinz Langer and Leonid Lerer. A panorama of modern operator theory and related topics. The Israel Gohberg memorial volume, Springer, pp.541-570, 2012, Operator Theory: Advances and Applications. 218, 978-3-0348-0220-8. ⟨10.1007/978-3-0348-0221-5_24⟩. ⟨hal-00725041⟩