Schatten Class Conditions for Functions of Schrödinger Operators

Abstract

We consider the difference f (H 1 ) − f (H 0 ), where H 0 = −∆ and H 1 = −∆ + V are the free and the perturbed Schrödinger operators in L 2 (R d ), and V is a real-valued short range potential. We give a sufficient condition for this difference to belong to a given Schatten class S p , depending on the rate of decay of the potential and on the smoothness of f (stated in terms of the membership in a Besov class). In particular, for p > 1 we allow for some unbounded functions f .

Kato smoothness and functions of perturbed self-adjoint operators

“…The latter operator is important in applications, which we develop in a separate paper [6]. In Appendix, we sketch the proof of Proposition 1.3 and of another technical statement of a similar nature.…”

“…We begin by defining DOI(a) for finite rank operators a. Let a be given by its Schmidt series,a = N n=1 s n (·, ϕ n )ψ n ,(4 6). where N is finite, {s n } are the singular values of a and {ϕ n }, {ψ n } are orthonormal sets.…”

“…Some applications. Here we briefly mention some applications of Theorems 1.1 and 1.2; these are developed in detail in the forthcoming publication [6].…”

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“…This relies on the origin being an interior point of Λ and may require an enlargement of L 0 , which can always be done. It follows that the right-hand side of (2.34) is bounded from above by some constant [11][12][13] to estimate differences of functions of the Laplacian and of a perturbation thereof.…”