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Trace formulas for Schrödinger operators with complex potentials on a half line

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Abstract

We consider Schrödinger operators with complex-valued decaying potentials on the half line. Such operator has essential spectrum on the half line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half line. We determine trace formula: sum of imaginary part of these eigenvalues plus some singular measure plus some integral from the Jost function. Moreover, we estimate sum of imaginary part of eigenvalues and singular measure in terms of the norm of potentials. In addition, we get bounds on the total number of eigenvalues, when the potential is compactly supported.

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Acknowledgements

Evgeny Korotyaev is grateful to Ari Laptev for discussions about the Schrödinger operators with complex potentials. He is also grateful to Alexei Alexandrov (St. Petersburg) for discussions and useful comments about Hardy spaces. Our study was supported by the RSF grant No 18-11-00032. Finally, we would like to thank the referees for thoughtful comments that helped us to improve the manuscript.

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  1. Department of Mathematical Analysis, Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia, 199034

    Evgeny Korotyaev

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  1. Evgeny Korotyaev
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Correspondence to Evgeny Korotyaev.

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Korotyaev, E. Trace formulas for Schrödinger operators with complex potentials on a half line. Lett Math Phys 110, 1–20 (2020). https://doi.org/10.1007/s11005-019-01210-x

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