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Limiting absorption principle and virtual levels of operators in Banach spaces

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Abstract

We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces.

Résumé

Nous passons en revue le principe d’absorption limite et sa relation avec les niveaux virtuels pour des opérateurs dans les espaces de Banach.

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Notes

  1. We suppose that in the twenties and thirties, between [41] and [75], the idea of the limiting absorption principle was being refined when V.S. Ignatowsky worked at St. Petersburg University, where in particular he taught mathematical theory of diffraction and likely was in contact with V.I. Smirnov. Let us mention that, besides his work on diffraction, Ignatowsky is known for his contributions to the theory of relativity (see [87]) and for developing optical devices while heading the theoretical division at GOMZ, the State Association for Optics and Mechanics (which later became known as “LOMO”). On 6 November 1941, during the blockade of St. Petersburg, Ignatowsky was arrested by NKVD (an earlier name of KGB), as a part of the “process of scientists”, and shot on 30 January 1942. (During this process, V.I. Smirnov was “credited” by NKVD the role of a Prime Minister in the government after the purportedly planned coup; Smirnov avoided the arrest because he was evacuated from St. Petersburg in August 1941, shortly before the blockade began.) As a result, Ignatowsky’s name has been unknown to many: the reference to his article disappeared from Smirnov’s “Course of higher mathematics” until post-1953 editions (see e.g. the English translation [76, §230]).

    Russians are used to such rewrites of the history, joking about the “History of the history of the Communist Party”, a reference to a mandatory and ever-changing Soviet-era ideological course in the first year of college. As the matter of fact, the very “Course of higher mathematics” mentioned above was started by V.I. Smirnov together with J.D. Tamarkin, with the first two volumes (published in 1924 and 1926) bearing both names, but after Tamarkin’s persecution by GPU (another earlier name of KGB) and his escape from the Soviet Union with smugglers over frozen lake Chudskoe in December 1924 [39], Tamarkin’s authorship eventually had to disappear. His coauthor Smirnov spent the next year pleading with the authorities (and succeeding!) for Tamarkin’s wife Helene Weichardt—who tried to follow her husband’s route with the smugglers over the icy lake but was intercepted at the border and jailed—to be released from prison and allowed to leave the Soviet Union to join her husband [7].

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Acknowledgements

The authors are most grateful to Gregory Berkolaiko, Kirill Cherednichenko, Fritz Gesztesy, Bill Johnson, Alexander V. Kiselev, Mark Malamud, Alexander Nazarov, Yehuda Pinchover, Roman Romanov, Thomas Schlumprecht, Vladimir Sloushch, Tatiana Suslina, Cyril Tintarev, Boris Vainberg, and Dmitrii Yafaev for their attention and advice. The authors are indebted to the anonymous referee for bringing to their attention several important references.

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  1. Laboratoire Mathématiques, Université Bourgogne Franche-Comté, 25030, Besançon CEDEX, France

    Nabile Boussaid

  2. Texas A&M University, College Station, TX, USA

    Andrew Comech

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Boussaid, N., Comech, A. Limiting absorption principle and virtual levels of operators in Banach spaces. Ann. Math. Québec 46, 161–180 (2022). https://doi.org/10.1007/s40316-021-00181-7

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