Some Applications of the Spectral Shift Operator.pdf

Some Applications of the Spectral Shift Operator.pdf

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Some Applications of the Spectral Shift Operator

a r X i v : m a t h / 9 9 0 3 1 8 6 v 1 [ m a t h .S P ] 3 1 M a r 1 9 9 9 SOME APPLICATIONS OF THE SPECTRAL SHIFT OPERATOR FRITZ GESZTESY AND KONSTANTIN A. MAKAROV Abstract. The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein’s trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators. 1. Introduction The concept of a spectral shift function, historically first introduced by I. M. Lif- shits [51], [52] and then developed into a powerful spectral theoretic tool by M. Krein [47], [48], [50], attracted considerable attention in the past due to its widespread applications in a variety of fields including scattering theory, relative index theory, spectral averaging and its application to localization properties of random Hamil- tonians, eigenvalue counting functions and spectral asymptotics, semi-classical ap- proximations, and trace formulas for one-dimensional Schro?dinger and Jacobi op- erators. For an extensive bibliography in this connection we refer to [33]; detailed reviews on the spectral shift function and its applications were published by Birman and Yafaev [11], [12] in 1993. The principal aim of this paper is to follow up on our recent paper [33], which was devoted to the introduction of a spectral shift operator Ξ(λ,H0, H) for a.e. λ ∈ R, associated with a pair of self-adjoint operators H0, H = H0 + V with V ∈ B1(H) (H a complex separable Hilbert space). In the special cases of sign- definite perturbations V ≥ 0 and V ≤ 0, Ξ(λ,H0, H) turns out to be a trace class operator inH, whose trace coincides with Krein’s spectral shift function ξ(λ,H0, H) for the pair (H0, H). While the special case V ≥ 0 has previously been studied by Carey [15], our aim in [33] was to trea

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