Trace Class Conditions for Functions of Schrödinger Operators

Abstract

Abstract:We consider the difference f (− + V )− f (− ) of functions of Schrödinger operators in L 2 (R d ) and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions f and in V belonging only to some L p space. This is motivated by applications in mathematical physics related to Lieb-Thirring inequalities. We show that in the particular case of Schrödinger operators the well-known sufficient conditions on f , based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems forOur key idea is the use of the limiting absorption principle.

Kato smoothness and functions of perturbed self-adjoint operators

“…In the p = 1 case, this is the result of our previous publication [5]. In the proof of Theorem 1.4, the concept of local S p -valued smoothness is important; in other words, one needs inclusions of the type GE H 0 (∆) ∈ Smooth p (H 0 ), where ∆ ⊂ (0, ∞).…”

“…In applications to mathematical physics (see e.g. [5]), one is often interested in functions f having a cusp-type singularity on the absolutely continuous spectrum and smooth elsewhere. It is also easy to reduce the question to functions f compactly supported on (0, ∞).…”

Schatten Class Conditions for Functions of Schrödinger Operators

“…Another generalization that we do not pursue here is to replace the pointwise assumption (1.2) on V by an integral assumption. In [5] we showed that this was possible for p = 1.…”

“…This paper is a continuation of [7], where this problem was considered in the general operator theoretic context. It is also a further development of [5], where the trace class membership of D(f ) was considered. As explained in [5] and briefly recalled in Subsection 1.6 below, this problem is in part motivated by applications to mathematical physics.…”

“…It can be shown that (see [15] [5], the sharpest sufficient conditions for Schatten class inclusions for D(f ) were obtained through general operator theoretic estimates of the form [13]…”