Abstract
The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.
Highlights
Let A and B be self-adjoint operators in a separable Hilbert space H and assume that the m-th powers of their resolvents differ by a trace class operator,(B − z IH)−m − ( A − z IH)−m ∈ S1(H), z ∈ ρ( A) ∩ ρ(B), (1.1)for some odd integer m ∈ N
For pairs of unitary operators and via Cayley transforms for the case m = 1 in (1.1) the spectral shift function and the trace formula were obtained later by Krein in [45]
We mention that the spectral shift function is closely connected with the scattering matrix via the famous Birman–Krein formula from [11,12]
Summary
Let A and B be self-adjoint operators in a separable Hilbert space H and assume that the m-th powers of their resolvents differ by a trace class operator,. The main objective of the present paper is to prove a representation formula for the spectral shift function in terms of an abstract Titchmarsh–Weyl m-function of two self-adjoint operators satisfying the condition (1.1), and to apply this result to different self-adjoint realizations of second-order elliptic PDEs and Schrödinger operators with compactly supported potentials. 5 we consider a formally symmetric uniformly elliptic second-order partial differential expression L with smooth coefficients on a bounded or unbounded domain in Rn, n ≥ 2, with compact boundary, and two self-adjoint realizations Aβ0 and Aβ1 of L subject to Robin boundary conditions βpγD f = γN f , where γD and γN denote the Dirichlet and Neumann trace operators, and βp ∈ C1(∂ ), p = 0, 1, are real-valued functions.
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