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Recent Developments in Spectral Theory for Non-self-adjoint Hamiltonians

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Mathematical Physics and Its Interactions (ICMPI 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 451))

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Abstract

The objective of this survey is to collect and elaborate on different tools, both well-established and more recent ones, which have been developed in the last decades to investigate spectral properties of non-self-adjoint operators of the form \(H=H_0+V\). More specifically, we will show how Hardy-type and Sobolev inequalities, together with Virial theorems and Birman-Schwinger principles enter into play in the analysis of the spectrum of these Hamiltonians.

Dedicated to Prof. Tohru Ozawa for his birthday, with deep gratitude for all he has done and sincere friendship.

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Notes

  1. 1.

    In classical quantum mechanics the notion of Hamiltonian is customarily connected to a purely self-adjoint setting (it is the observable which represents the total energy of the system). Nevertheless, it turned out that adopting a less fundamental approach one can provide a relaxation of the notion which is meaningful also for non-self-adjoint operators (see the introduction in [48] and references therein.).

  2. 2.

    Actually, this is just one of the many possible definition for the essential spectrum, which are all equivalent only in the self-adjoint setting. In particular, the one we are using here is \(\sigma _{e_3}\) in the monograph [31] by Edmunds and Evans.

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Acknowledgements

The research of the first author (L.C.) is Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477 - SFB 1173. The third author (N.M.S.) is supported by the EXPRO grant No. 20-17749X of the Czech Science Foundation. He is also member of the Gruppo Nazionale per L’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. Fakultät Für Mathematik, Institut Für Analysis, Karlsruher Institut Für Technologie (KIT), Englerstraße 2, 76131, Karlsruhe, Germany

    Lucrezia Cossetti

  2. Ikerbasque & Departamento de Matematicas, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Aptdo. 644, 48080, Bilbao, Spain

    Luca Fanelli

  3. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000, Prague 2, Czechia

    Nico M. Schiavone

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  1. Lucrezia Cossetti
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  2. Luca Fanelli
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Correspondence to Luca Fanelli .

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  1. Department of Mathematics, Saitama University, Saitama, Japan

    Shuji Machihara

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Cossetti, L., Fanelli, L., Schiavone, N.M. (2024). Recent Developments in Spectral Theory for Non-self-adjoint Hamiltonians. In: Machihara, S. (eds) Mathematical Physics and Its Interactions. ICMPI 2021. Springer Proceedings in Mathematics & Statistics, vol 451. Springer, Singapore. https://doi.org/10.1007/978-981-97-0364-7_8

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