The Spectral Density of a Difference of Spectral Projections

Abstract

Abstract. Let H 0 and H be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if λ belongs to the absolutely continuous spectrum of H 0 and H, then the difference of spectral projectionsin general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations D ε (λ) of D(λ), given bywhere ψ ε (x) = ψ(x/ε) and ψ(x) is a smooth real-valued function which tends to ∓1/2 as x → ±∞. We prove that the eigenvalues of D ε (λ) concentrate to the absolutely continuous spectrum of D(λ) as ε → +0. We show that the rate of concentration is proportional to | log ε| and give an explicit formula for the asymptotic density of these eigenvalues. It turns out that this density is independent of ψ. The proof relies on the analysis of Hankel operators.

Schatten Class Conditions for Functions of Schrödinger Operators

“…Yet another example is the discrete Laplacian. We omit the details, but refer to Section 11 of [14] for some of the necessary ingredients for these extensions in some cases.…”

“…The discontinuous limiting function is f (x) = χ {x<µ} , while the functions approximating this function can be chosen smooth; see Section 3 in [8]. To be more precise, in this problem the product of f (H 1 ) and f (H 0 ) rather than their difference appears, but a mathematically closely related problem for the difference was studied by one of us in [14]. In fact, in view of the latter work we believe that for both the operator norm and the Schatten norm with any fixed 0 < p < ∞ the assumptions on ρ and f in Theorems 1.1 and 1.2 are best possible.…”