{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T20:05:56Z","timestamp":1790798756428,"version":"4.1.0"},"reference-count":52,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007648","name":"Durban University of Technology","doi-asserted-by":"publisher","award":["x"],"award-info":[{"award-number":["x"]}],"id":[{"id":"10.13039\/100007648","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Non-Hermitian quantum-Hamiltonian-candidate combination     H \u03bb     of a non-Hermitian unperturbed operator     H =  H 0      with an arbitrary \u201csmall\u201d non-Hermitian perturbation     \u03bb W     is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh\u2013Schr\u00f6dinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed.<\/jats:p>","DOI":"10.3390\/e22010080","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T04:06:51Z","timestamp":1578629211000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Theory of Response to Perturbations in Non-Hermitian Systems Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6076-0093","authenticated-orcid":false,"given":"Miloslav","family":"Znojil","sequence":"first","affiliation":[{"name":"Institute of System Science, Durban University of Technology, P. O. Box 1334, Durban 4000, South Africa"},{"name":"The Czech Academy of Sciences, Nuclear Physics Institute, Hlavn\u00ed 130, 250 68 \u0158e\u017e, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kato, T. (1966). Perturbation Theory for Linear Operators, Springer.","DOI":"10.1007\/978-3-642-53393-8"},{"key":"ref_2","unstructured":"Messiah, A. (1961). Quantum Mechanics, North Holland."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2307\/1968538","article-title":"On one-parameter unitary groups in Hilbert Space","volume":"33","author":"Stone","year":"1932","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5243","DOI":"10.1103\/PhysRevLett.80.5243","article-title":"Real spectra in non-Hermitian Hamiltonians having P T symmetry","volume":"80","author":"Bender","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1103\/PhysRev.102.1217","article-title":"General Theory of Spin-Wave Interactions","volume":"102","author":"Dyson","year":"1956","journal-title":"Phys. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","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","author":"Scholtz","year":"1992","journal-title":"Ann. Phys. (NY)"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1088\/0034-4885\/70\/6\/R03","article-title":"Making sense of non-Hermitian Hamiltonians","volume":"70","author":"Bender","year":"2007","journal-title":"Rep. Prog. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1142\/S0219887810004816","article-title":"Pseudo-Hermitian Quantum Mechanics","volume":"7","author":"Mostafazadeh","year":"2010","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Bagarello, F., Gazeau, J.-P., Szafraniec, F., and Znojil, M. (2015). Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley.","DOI":"10.1002\/9781118855300"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"85","DOI":"10.14311\/AP.2014.54.0085","article-title":"The coupled-cluster approach to quantum many-body problem in a three-Hilbert-space reinterpretation","volume":"54","author":"Bishop","year":"2014","journal-title":"Acta Polytech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"244026","DOI":"10.1088\/1751-8113\/41\/24\/244026","article-title":"Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians","volume":"41","author":"Smilga","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.aop.2017.08.009","article-title":"Non-Hermitian interaction representation and its use in relativistic quantum mechanics","volume":"385","author":"Znojil","year":"2017","journal-title":"Ann. Phys. (NY)"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"032124","DOI":"10.1103\/PhysRevA.100.032124","article-title":"Unitarity corridors to exceptional points","volume":"100","author":"Znojil","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_14","first-page":"001","article-title":"Three-Hilbert-space formulation of Quantum Mechanics","volume":"5","author":"Znojil","year":"2009","journal-title":"Symm. Integ. Geom. Meth. Appl. SIGMA"},{"key":"ref_15","first-page":"241","article-title":"C*-algebra techniques in numerical analysis","volume":"35","author":"Roch","year":"1996","journal-title":"J. Oper. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103513","DOI":"10.1063\/1.4934378","article-title":"Pseudospectra in non-Hermitian quantum mechanics","volume":"56","author":"Siegl","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1112\/S0024609302001248","article-title":"Non-self-adjoint differential operators","volume":"34","author":"Davies","year":"2002","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Trefethen, L.N., and Embree, M. (2005). Spectra and Pseudospectra, Princeton University Press.","DOI":"10.1515\/9780691213101"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4256","DOI":"10.1063\/1.1727484","article-title":"On the correlation problem in atomic and molecular systems. Calculation of wavefunction components in Ursell-type expansion using quantum field theoretical methods","volume":"45","year":"1966","journal-title":"J. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0375-9474(71)90122-9","article-title":"Boson description of collective states:(I). Derivation of the boson transformation for even fermion systems","volume":"172","author":"Janssen","year":"1971","journal-title":"Nucl. Phys. A"},{"key":"ref_21","unstructured":"Dieudonne, J. (1961). Quasi-Hermitian Operators. Proc. Int. Symp. Lin. Spaces, Pergamon."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Antoine, J.-P., and Trapani, C. (2015). Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces. Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley.","DOI":"10.1002\/9781118855300.ch7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"012007","DOI":"10.1088\/1742-6596\/1194\/1\/012007","article-title":"Beyond Hilbert space: RHS, PIP and all that","volume":"1194","author":"Antoine","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Znojil, M. (2015). Non-Selfadjoint Operators in Quantum Physics: Ideas, People, Trends. Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley.","DOI":"10.1002\/9781118855300.ch1"},{"key":"ref_25","unstructured":"Wilkinson, J.H. (1965). The Algebraic Eigenvalue Problem, Oxford University Press."},{"key":"ref_26","unstructured":"Znojil, M. (2001). Conservation of pseudo-norm in PT symmetric quantum mechanics. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"042122","DOI":"10.1103\/PhysRevA.95.042122","article-title":"Problem of the coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics","volume":"95","author":"Znojil","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_28","first-page":"20180264","article-title":"The minimally anisotropic metric operator in quasi-hermitian quantum mechanics","volume":"474","author":"Lotoreichik","year":"2018","journal-title":"Proc. Roy. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"462","DOI":"10.14311\/AP.2017.57.0462","article-title":"Crypto-Hermitian Approach to the Klein\u2013Gordon Equation","volume":"57","year":"2017","journal-title":"Acta Polytech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"9557","DOI":"10.1088\/0305-4470\/37\/40\/016","article-title":"Relativistic supersymmetric quantum mechanics based on Klein\u2013Gordon equation (hep-th\/0408232)","volume":"37","author":"Znojil","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_31","first-page":"001","article-title":"On the role of the normalization factors kappa n and of the pseudo-metric P in crypto-Hermitian quantum models","volume":"4","author":"Znojil","year":"2008","journal-title":"SIGMA"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012120","DOI":"10.1088\/1742-6596\/1194\/1\/012120","article-title":"Nonlinearity of perturbations in-symmetric quantum mechanics","volume":"1194","author":"Znojil","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Bender, C.M. (2018). PT Symmetry in Quantum and Classical Physics, World Scientific.","DOI":"10.1142\/q0178"},{"key":"ref_34","first-page":"10171","article-title":"Metric operator in pseudo-Hermitian quantum mechanics and the imaginary cubic potential","volume":"39","author":"Mostafazadeh","year":"2006","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"244012","DOI":"10.1088\/1751-8113\/41\/24\/244012","article-title":"Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem","volume":"41","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"032114","DOI":"10.1103\/PhysRevA.97.032114","article-title":"Admissible perturbations and false instabilities in PT-symmetric quantum systems","volume":"97","author":"Znojil","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4863","DOI":"10.1088\/1751-8113\/40\/18\/012","article-title":"Maximal couplings in PT-symmetric chain-models with the real spectrum of energies","volume":"40","author":"Znojil","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"13131","DOI":"10.1088\/1751-8113\/40\/43\/020","article-title":"Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime","volume":"40","author":"Znojil","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"eaar7709","DOI":"10.1126\/science.aar7709","article-title":"Exceptional points in optics and photonics","volume":"363","author":"Miri","year":"2019","journal-title":"Science"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/0003-4916(58)90007-1","article-title":"Unified theory of nuclear reactions","volume":"5","author":"Feshbach","year":"1958","journal-title":"Ann. Phys. (NY)"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Moiseyev, N. (2011). Non-Hermitian Quantum Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511976186"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.physleta.2004.04.036","article-title":"Linear representation of energy-dependent Hamiltonians (quant-ph\/0403223)","volume":"326","author":"Znojil","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"R3255","DOI":"10.1103\/PhysRevD.55.R3255","article-title":"Nonperturbative Calculation of Symmetry Breaking in Quantum Field Theory","volume":"55","author":"Bender","year":"1997","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/S0375-9601(99)00429-6","article-title":"PT symmetric harmonic oscillators","volume":"259","author":"Znojil","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/S0370-2693(00)00569-4","article-title":"Supersymmetry without hermiticity within PT symmetric quantum mechanics (hep-th\/0003277)","volume":"483","author":"Znojil","year":"2000","journal-title":"Phys. Lett. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"7165","DOI":"10.1088\/0305-4470\/33\/40\/313","article-title":"Systematic search for PT symmetric potentials with real energy spectra","volume":"33","author":"Znojil","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"255206","DOI":"10.1088\/1751-8113\/41\/25\/255206","article-title":"A non-Hermitian PT symmetric Bose-Hubbard model: Eigenvalue rings from unfolding higherorder exceptional points","volume":"41","author":"Graefe","year":"2008","journal-title":"J. Phys. A"},{"key":"ref_48","unstructured":"G\u00fcnther, U., and Stefani, F. (2019). IR-truncated PT-symmetric ix3 model and its asymptotic spectral scaling graph. arXiv."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Bagarello, F. (2015). Deformed Caqnonical (Anti-)Commutation Relations adn Non-Self-Adjoint Hamiltonians. Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, Wiley.","DOI":"10.1002\/9781118855300"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"121702","DOI":"10.1103\/PhysRevD.86.121702","article-title":"On the metric operator for the imaginary cubic oscillator","volume":"86","author":"Siegl","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"023415","DOI":"10.1103\/PhysRevA.84.023415","article-title":"Shortcuts to adiabaticity for non-Hermitian systems","volume":"84","author":"Chen","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"444016","DOI":"10.1088\/1751-8113\/45\/44\/444016","article-title":"The physics of exceptional points","volume":"45","author":"Heiss","year":"2012","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/80\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:42:35Z","timestamp":1760362955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/80"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010080"],"URL":"https:\/\/doi.org\/10.3390\/e22010080","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,9]]}}}