F. Gesztesy, K. A. Makarov, and A. K. Motovilov
Monotonicity and Concavity Properties of The Spectral Shift Function
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ABSTRACT.  Let H_0 and V(s) be self-adjoint, V, V' continuously differentiable in trace norm with V''(s)\geq 0 for s\in (s_1,s_2), and denote by
{E_{H(s)}(\lambda)}_{\lambda\in\bbR} the family of spectral projections of H(s)=H_0+V(s). Then we prove for given \mu\in\bbR, that s\longmapsto
\tr (V'(s)E_{H(s)}((-\infty, \mu))) is a nonincreasing function with respect to s, extending a result of Birman and Solomyak. Moreover, denoting by
\zeta (\mu,s)=\int_{-\infty}^\mu d\lambda \xi(\lambda,H_0,H(s)) the integrated spectral shift function for the pair (H_0,H(s)), we prove concavity of
\zeta (\mu,s) with respect to s, extending previous results by Geisler, Kostrykin, and Schrader. Our proofs employ operator-valued Herglotz functions and establish the latter as an effective tool in this context.