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Absence of embedded eigenvalues for non-local Schrödinger operators

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Abstract

We consider non-local Schrödinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman–Schwinger principle, previously developed for classical Schrödinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.

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Acknowledgements

AI thanks JSPS KAKENHI (Grant Numbers JP20K03625, JP21K03279 and JP21KK0245) and Tokyo University of Science Grant for International Joint Research for support. JL gratefully thanks the hospitality of IHES, Bures-sur-Yvette, and both thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support (EPSRC grant no EP/K032208/1) and hospitality during the programme “Fractional Differential Equations” where work on this paper was undertaken. IS thanks JSPS KAKENHI (Grant Numbers JP16 K17612 and JP20K03628) for support. The authors thank Professor Masahito Ohta of Tokyo University of Science for his valuable comments and also thank reviewers.

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Authors and Affiliations

  1. Katsushika Division, Institute of Arts and Sciences, Tokyo University of Science, Tokyo, 125-8585, Japan

    Atsuhide Ishida

  2. Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053, Budapest, Hungary

    József Lőrinczi

  3. Department of Mathematics, Shinshu University, Matsumoto 3-1-1, Asahi, Japan

    Itaru Sasaki

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  1. Atsuhide Ishida
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  2. József Lőrinczi
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  3. Itaru Sasaki
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Correspondence to Atsuhide Ishida.

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Ishida, A., Lőrinczi, J. & Sasaki, I. Absence of embedded eigenvalues for non-local Schrödinger operators. J. Evol. Equ. 22, 82 (2022). https://doi.org/10.1007/s00028-022-00836-0

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