Concavity of Eigenvalue Sums and the Spectral Shift Function

Abstract

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from $-\infty$ to $\lambda$ (from $\lambda$ to $+\infty$) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered

Bounds on the Spectral Shift Function and the Density of States

“…[59], and the study of surface potentials, see [10,36]. Various of its properties are discussed in the literature: monotonicity and concavity in [20,22,35], the asymptotic behaviour in the large coupling constant [43,46,44] and semiclassical limit [42,40]. See [5,34] for surveys.…”

Graph Subspaces and the Spectral Shift Function

“…For many more references the interested reader can consult [23], [24], [26], [27], [58], [60]. For recent results we refer to [25], [35], [59], [61], [62], and [64].…”

Regularity of the Surface Density of States

“…A detailed account on the theory of the spectral shift function can be found in the review [4] and in the book [25]. For recent studies we refer to [14,10] and references therein. Recently the spectral shift function found a number of applications in the theory of random Schrödinger operators [15], [16], [17], [5], [6], [23], [18], [8].…”