Quasi-Hermitian operators in quantum mechanics and the variational principle
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Publication:1191712
DOI10.1016/0003-4916(92)90284-SzbMath0749.47041OpenAlexW2084732260MaRDI QIDQ1191712
F. J. W. Hahne, Frederik G. Scholtz, Hendrik B. Geyer
Publication date: 27 September 1992
Published in: Annals of Physics (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1016/0003-4916(92)90284-s
variational principleconsistent quantum mechanical systemnon-Hermitian observablesquantum-mechanical interpretationset of non-Hermitian operators
Applications of operator theory in the physical sciences (47N50) Linear symmetric and selfadjoint operators (unbounded) (47B25) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10)
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Cites Work