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