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Some Applications of the Spectral Shift Operator
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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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