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Self-adjoint extensions of bipartite Hamiltonians

Part of: Miscellaneous applications of functional analysis General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  22 June 2021

Daniel Lenz ,
Timon Weinmann  and
Melchior Wirth
Daniel Lenz
Affiliation:
Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany (daniel.lenz@uni-jena.de)
Timon Weinmann
Affiliation:
Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia (st082214@student.spbu.ru)
Melchior Wirth
Affiliation:
Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany (melchior.wirth@ist.ac.at)
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Abstract

We compute the deficiency spaces of operators of the form $H_A{\hat {\otimes }} I + I{\hat {\otimes }} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor.47(38) (2014) 385301], but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.

Keywords

operator theoryfunctional analysismathematical physicsquantum systemsspectral theoryself-adjoint extensions

MSC classification

Primary: 46N50: Applications in quantum physics
Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 433 - 447
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

*

Address at the time of publication: IST Austria, Klosterneuburg, Austria.

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