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SIGMA 7 (2011), 112, 16 pages arXiv:1107.5916
https://doi.org/10.3842/SIGMA.2011.112
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”
Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
Andrey V. Sokolov
V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University, 198504 St. Petersburg, Russia
Received August 06, 2011, in final form November 25, 2011; Published online December 05, 2011
Abstract
This part is a continuation of the Part I where we built
resolutions of identity for certain non-Hermitian Hamiltonians
constructed of biorthogonal sets of their eigen- and associated
functions for the spectral problem defined on entire axis.
Non-Hermitian Hamiltonians under consideration are taken with
continuous spectrum and the following cases are examined: an
exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and
an exceptional point situated inside of continuous spectrum. In the
present work the rigorous proofs are given for the resolutions of
identity in both cases.
Key words:
non-Hermitian quantum mechanics; supersymmetry; exceptional points; resolution of identity.
pdf (345 kb)
tex (13 kb)
References
- Andrianov A.A., Sokolov A.V.,
Resolutions of identity for some non-Hermitian Hamiltonians. I. Exceptional point in continuous spectrum,
SIGMA 7 (2011), 111, 19 pages,
arXiv:1107.5911.
- Gel'fand I.M., Vilenkin N.J., Generalized functions, Vol. 4, Some
applications of harmonic analysis, Academic Press, New York, 1964.
- Sokolov A.V., Andrianov A.A., Cannata F.,
Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states,
J. Phys. A: Math. Gen. 39 (2006), 10207-10227,
quant-ph/0602207.
- Andrianov A.A., Cannata F., Sokolov A.V.,
Spectral singularities for non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity,
J. Math. Phys. 51 (2010), 052104, 22 pages,
arXiv:1002.0742.
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