{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T02:24:53Z","timestamp":1772331893935,"version":"3.50.1"},"reference-count":22,"publisher":"AIP Publishing","issue":"1","funder":[{"name":"PEst- C\/MAT 2011","award":["UI0324"],"award-info":[{"award-number":["UI0324"]}]},{"name":"UID\/FIS 2016","award":["04564"],"award-info":[{"award-number":["04564"]}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:p>In this article, we review the quantum mechanical setting associated with a non-self-adjoint Hamiltonian with a real spectrum. The spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non-self-adjoint operators with implications in quantum mechanics are discussed.<\/jats:p>","DOI":"10.1063\/1.5048577","type":"journal-article","created":{"date-parts":[[2019,1,8]],"date-time":"2019-01-08T17:04:07Z","timestamp":1546967047000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Non-self-adjoint operators with real spectra and extensions of quantum mechanics"],"prefix":"10.1063","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2600-6489","authenticated-orcid":false,"given":"N.","family":"Bebiano","sequence":"first","affiliation":[{"name":"Department of Mathematics, CMUC, University of Coimbra 1 , P 3001-454 Coimbra, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2614-7775","authenticated-orcid":false,"given":"J.","family":"da Provid\u00eancia","sequence":"additional","affiliation":[{"name":"Department of Physics, CFisUC, University of Coimbra 2 , P 3004-516 Coimbra, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2019,1,8]]},"reference":[{"key":"2023070204383933800_c1","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1007\/s13538-011-0010-9","article-title":"Non Hermitian operators with real spectra in quantum mechanics","volume":"41","year":"2011","journal-title":"Braz. J. Phys."},{"key":"2023070204383933800_c2","doi-asserted-by":"publisher","first-page":"023531","DOI":"10.1063\/1.3300804","article-title":"Construction of pseudo-bosons systems","volume":"51","year":"2010","journal-title":"J. Math. Phys."},{"key":"2023070204383933800_c3","doi-asserted-by":"publisher","first-page":"4110","DOI":"10.1007\/s10773-014-2487-9","article-title":"D-deformed harmonic oscillators","volume":"54","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"2023070204383933800_c4","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.aop.2015.02.034","article-title":"Some results on the dynamics and transition probabilities for non self-adjoint Hamiltonians","volume":"356","year":"2015","journal-title":"Ann. Phys."},{"key":"2023070204383933800_c5","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.aop.2015.08.010","article-title":"Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces","volume":"362","year":"2015","journal-title":"Ann. Phys."},{"key":"2023070204383933800_c6","volume-title":"Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects","year":"2015"},{"key":"2023070204383933800_c7","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/0003-4916(92)90284-s","article-title":"Quasi-Hermitian operators in quantum mechanics and the variational principle","volume":"213","year":"1992","journal-title":"Ann. Phys."},{"key":"2023070204383933800_c8","doi-asserted-by":"publisher","first-page":"5243","DOI":"10.1103\/physrevlett.80.5243","article-title":"Real spectra in non-Hermitian Hamiltonians having PT symmetry","volume":"80","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"2023070204383933800_c9","doi-asserted-by":"publisher","first-page":"270401","DOI":"10.1103\/physrevlett.89.270401","article-title":"Complex extension of quantum mechanics","volume":"89","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"2023070204383933800_c10","volume-title":"Quantum Mechanics","year":"1965"},{"key":"2023070204383933800_c11"},{"key":"2023070204383933800_c12","first-page":"115","article-title":"Quasi-Hermitian operators","year":"1961"},{"key":"2023070204383933800_c13","doi-asserted-by":"publisher","first-page":"3715","DOI":"10.1088\/0305-4470\/39\/14\/014","article-title":"Nonlinear pseudo-supersymmetry in the framework of N-fold supersymmetry","volume":"39","year":"2006","journal-title":"J. Phys. A: Math. Gen."},{"key":"2023070204383933800_c14","volume-title":"Classes of Linear Operators","year":"1990"},{"key":"2023070204383933800_c15","doi-asserted-by":"crossref","DOI":"10.1002\/9781118855300","article-title":"Elements of spectral theory without the spectral theorem","volume-title":"Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects","author":"Bagarello","year":"2015"},{"key":"2023070204383933800_c16","doi-asserted-by":"publisher","first-page":"103513","DOI":"10.1063\/1.4934378","article-title":"Pseudospectra in non-Hermitian quantum mechanics","volume":"56","year":"2015","journal-title":"J. Math. Phys."},{"key":"2023070204383933800_c17","first-page":"201","article-title":"Geometry and the integer quantum Hall effect","volume-title":"Geometrical Analysis and the Lie Theory, in Mathematics and Physics","author":"Carey","year":"1998"},{"key":"2023070204383933800_c18","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1142\/s0219887810004816","article-title":"Pseudo-Hermitian representation of quantum mechanics","volume":"7","year":"2010","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"2023070204383933800_c19","doi-asserted-by":"publisher","first-page":"7081","DOI":"10.1088\/0305-4470\/36\/25\/312","article-title":"Exact PT-symmetry is equivalent to Hermiticity","volume":"36","year":"2003","journal-title":"J. Phys. A: Math. Gen."},{"key":"2023070204383933800_c20","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1063\/1.1640796","article-title":"Transition elements for a non-Hermitian quadratic Hamiltonian","volume":"45","year":"2004","journal-title":"J. Math. Phys."},{"key":"2023070204383933800_c21","doi-asserted-by":"publisher","first-page":"121702(R)","DOI":"10.1103\/physrevd.86.121702","article-title":"The metric operator for the imaginary cubic oscillator","volume":"86","year":"2012","journal-title":"Phys. Rev. D"},{"key":"2023070204383933800_c22","doi-asserted-by":"publisher","first-page":"122","DOI":"10.2991\/jnmp.2002.9.s2.11","article-title":"Should PT symmetric quantum mechanics Be interpreted as nonlinear?","volume":"9","year":"2002","journal-title":"J. Nonlinear Math. Phys."}],"container-title":["Journal of Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.5048577\/13509621\/012104_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.5048577\/13509621\/012104_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,2]],"date-time":"2023-07-02T04:38:49Z","timestamp":1688272729000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jmp\/article\/60\/1\/012104\/318628\/Non-self-adjoint-operators-with-real-spectra-and"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1,1]]}},"URL":"https:\/\/doi.org\/10.1063\/1.5048577","relation":{},"ISSN":["0022-2488","1089-7658"],"issn-type":[{"value":"0022-2488","type":"print"},{"value":"1089-7658","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2019,1]]},"published":{"date-parts":[[2019,1,1]]},"article-number":"012104"}}