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Analysis and Operator Theory pp 149–165Cite as

On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 146))

Abstract

The aim of this paper is to review and compare the spectral properties of the Schrödinger operators \(-\varDelta + U\) (\(U\ge 0\)) and \(-\varDelta + i V\) in \(L^2(\mathbb R^d)\) for \(C^\infty \) real potentials U or V with polynomial behavior. The case with magnetic field will be also considered. We present the existing criteria for essential self-adjointness, maximal accretivity, compactness of the resolvent, and maximal inequalities. Motivated by recent works with X. Pan, Y. Almog, and D. Grebenkov, we actually improve the known results in the case with purely imaginary potential.

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Notes

  1. 1.

    The results of this paper have been presented by the first author at the Kato centennial conference in Tokyo in September 2017.

  2. 2.

    There are actually two different proofs proposed by J. Nourrigat a rather direct one and another based to the analysis of \(\sum _j \check{X}_j^2 + i \check{X}_0\) the difficulty (but this was sometimes treated in [15]) that this operator is no more hypoelliptic.

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Acknowledgements

The authors would like to thank J. Camus for the transmission of [11] and Z. Shen for an enlightening clarification [34]. The first author would like to thank S. Nakamura and K. Yajima for their invitation to the Kato centennial conference and the RIMS Kyoto for its support.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, CNRS and Université de Nantes, 2 rue de la Houssinière, 44322, Nantes Cedex, France

    Bernard Helffer

  2. LMR EA 4535 and FR CNRS 3399, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039 51687, REIMS Cedex 2, France

    Jean Nourrigat

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  1. Bernard Helffer
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Correspondence to Bernard Helffer .

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Editors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Aix-Marseille University, Marseille, France

    Valentin A. Zagrebnov

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Helffer, B., Nourrigat, J. (2019). On the Domain of a Magnetic Schrödinger Operator with Complex Electric Potential. In: Rassias, T.M., Zagrebnov, V.A. (eds) Analysis and Operator Theory . Springer Optimization and Its Applications, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-030-12661-2_8

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