Monotonicity and Concavity Properties of The Spectral Shift Function

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\big (V'(s)E_{H(s)}((-\infty, \mu))\big) $ 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.