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Schatten Class Conditions for Functions of Schrödinger Operators

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Abstract

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.

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Acknowledgements

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. Mathematisches Institut, Ludwig-Maximilians Universität München, Theresienstr. 39, 80333, Munich, Germany

    Rupert L. Frank

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Rupert L. Frank

  3. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    Alexander Pushnitski

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  1. Rupert L. Frank
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  2. Alexander Pushnitski
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Correspondence to Rupert L. Frank.

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Communicated by Jan Derezinski.

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Frank, R.L., Pushnitski, A. Schatten Class Conditions for Functions of Schrödinger Operators. Ann. Henri Poincaré 20, 3543–3562 (2019). https://doi.org/10.1007/s00023-019-00838-8

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