Annales Henri Poincaré

, Volume 20, Issue 11, pp 3543–3562 | Cite as

Schatten Class Conditions for Functions of Schrödinger Operators

  • Rupert L. FrankEmail author
  • Alexander Pushnitski


We consider the difference \(f(H_1)-f(H_0)\), where \(H_0=-\Delta \) and \(H_1=-\Delta +V\) are the free and the perturbed Schrödinger operators in \(L^2({{\mathbb {R}}}^d)\), respectively, in which V is a real-valued short range potential. We give a sufficient condition for this difference to belong to a given Schatten class \({\mathbf {S}}_p\), 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.



Partial support by U.S. National Science Foundation DMS-1363432 (R.L.F.) is acknowledged. A.P. is grateful to Caltech for hospitality.


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Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians Universität MünchenMunichGermany
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of MathematicsKing’s College LondonLondonUK

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