Inverted oscillator: pseudo hermiticity and coherent states


  • Rahma Zerimeche
  • Rostom Moufok
  • Nadjat Amaouche
  • Mustapha Maamache Universite Ferhat Abbas Setif 19000



Inverted harmonic oscillator; harmonic Hamiltonian


It is known that the standard and the inverted harmonic oscillator are different. Replacing thus ω by ±iω in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (∓iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent states which minimize the quantum mechanical uncertainty between the position and the momentum. This paper is dedicated to the memory of Omar Djemli and Nouredinne Mebarki who died due to covid 19.



G. Barton: Quantum mechanics of the inverted oscillator potential, Ann. Phys. 166 (1986) 322,

S. Baskoutas, A. Jannussis, R. Mignani and V Papatheou: Tunnelling process for non-Hermitian systems: the complexfrequency inverted oscillator, J. Phys. A: Math. Gen. 17 (1993) L819,

R. K. Bhaduri, A. Khare and J. Law: Phase of the Riemann Zeta function and the inverted harmonic oscillator, Phys. Rev. E. 52 (1995) 486,

R. K. Bhaduri, A. Khare, S. M. Reimann and E. L. Tomusiakl: The Riemann Zeta Function and the Inverted Harmonic Oscillator, Ann. Physics. 254 (1997) 25,

Mario Castagnino, Roberto Diener, Luis Lara,and Gabriel Puccin: Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator, Int. J. of Theor. Phys. 36 (1997) 2349,

T. Shimbori: Operator methods of the parabolic potential barrier, Phys. Lett. A. 273 (2000) 37,

I. A. Pedrosa, I. Guedes: Quantum States of a Generalized Time-Dependent Inverted Harmonic Oscillator, Int. J. Mod. Phys. B. 18 (2004) 1379,

D. Chruscinski: Quantum mechanics of damped systems. II. Damping and parabolic potential barrier, J. Math. Phys. 45 (2004) 841,

D. Chruscinski: Quantum damped oscillator II: Bateman’s Hamiltonian vs. 2D parabolic potential barrier, Ann Phys. 321 (2006) 840,

C. Yuce, A. Kilic and A. Coruh: Inverted Oscillator, Phys. Phys. Scr. 74 (2006) 114,

C. A. Muntoz, J. Rueda-Paz, and K. B. Wolf: Discrete repulsive oscillator wave functions, J. Phys. A. 42 (2009) 485210,

D. Bermudez and D. J. Fernandez C: Factorization method and new potentials from the inverted oscillator, Ann. Phys. 333 (2013) 290,

M. Maamache, Y. Bouguerra and J. R. Choi: Time behavior of a Gaussian wave packet accompanying the generalized coherent state for the inverted oscillator, Prog. Theor. Exp. Phys. 063A01, (2016),

K. Rajeev, S. Chakraborty and T. Padmanabhan: Inverting a normal harmonic oscillator: physical interpretation and applications, Gen. Relativ. Gravit. 50 (2018) 116,

R. D. Mota, D.Ojeda-Guillen, M. Salazar-Ramırez, and V. D. Granados, Non-Hermitian inverted harmonic oscillator-type Hamiltonians generated fromsupersymmetry with reflections, Mod. Phys. Lett. A 34 (2019) 1950028,

T. Shimbori and T. Kobayashi: Complex eigenvalues of the parabolic potential barrier and Gel’fand triplet, Nuovo Cimento B. 115 (2000) 325,

K. Aouda, Naohiro Kanda, Shigefumi Naka and Haruki Toyoda: Ladder operators in repulsive harmonic oscillator with application to the Schwinger effect, Phys. Rev. D. 102 (2020) 025002,

A. Bhattacharyya, W. Chemissany, S. S. Haque, J. Murugan and B. Yan: The multi-faceted inverted harmonic oscillator: Chaos and complexity, SciPost Phys. Core 4 (2021) 002,

G. Chong and W. Hai: Dynamical evolutions of matterwave bright solitons in an inverted parabolic potential, J. Phys. B. 40 (2007) 211,

G. Felder, A. Frolov, L. Kofman, and A. Linde: Cosmology with negative potentials, Phys. Rev. D. 66 (2002) 023507,

S. Tarzi, The inverted harmonic oscillator: some statistical properties, J. Phys. A: Math. Gen. 21 (1988) 3105,

C. M. Bender and S. Boettcher: Real Spectra in NonHermitian Hamiltonians Having PT -Symmetry, Phys. Rev. Lett. 80 (1998) 5243,

C. M. Bender, Dorje C. Brody, and Hugh F. Jones: Complex Extension of Quantum Mechanics, Phys. Rev. Lett. 89 (2002) 270401,

A. Mostafazadeh: Pseudo-Hermiticity versus PT -symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 205 (2002) 214,

L. Ge and H. E. Tureci: Antisymmetric PT -photonic structures with balanced positive- and negative-index mateials, Phys. Rev. A. 88 (2013) 053810,

J.H. Wu, M. Artoni, and G. C. La Rocca: Non-Hermitian Degeneracies and Unidirectional Reflectionless Atomic Lattices, Phys. Rev. Lett. 113 (2014) 123004,

S. Longhi: Phase transitions in Wick-rotated PT -symmetric optics, Ann. Phys. 360 (2015) 150,

P. Peng, W. Cao, C.Shen, W. Qu , J.Wen, L.Jiang and Y.Xiao: Anti-parity–time symmetry with flying atoms, Nat. Phys. 12 (2016) 1139,

M. Maamache and L. Kheniche: Anti-PT symmetry for a non-Hermitian Hamiltonian, Prog. Theor. Exp. Phys. 123A01, (2020),

A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach To Quantum Optics, John Wiley & Sons Inc. (2009).

E. Schrödinger: Der stetige Ubergang von der Mikro-zur Makromechanik, Naturwissenschaften 14 (1926) 664.

R. J. Glauber: The Quantum Theory of Optical Coherence, Phys. Rev.130 (1963) 2529, Photon Correlations, Phys. Rev. Lett. 10 (1963) 84, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766,

J. R. Klauder: Continuous-Representation Theory. I. Postulates of Continuous-Representation Theory, J. Math. Phys. 4(1963)1055, Continuous-Representation Theory. II. Generalized Relation between Quantum and Classical Dynamics, J. Math. Phys. 4 (1963) 1058,

E. C. G. Sudarshan: Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, Phys. Rev. Lett. 10 (1963) 277,

N. Amaouche, I. Bouguerche, R. Zerimeche and M. Maamache, Anti-PT symmetric harmonic oscillator and its relation to the inverted harmonic oscillator, arXiv:2204.10780v1 (2022), 10780.




How to Cite

R. Zerimeche, R. Moufok, N. Amaouche, and M. Maamache, “Inverted oscillator: pseudo hermiticity and coherent states ”, Rev. Mex. Fís., vol. 69, no. 1 Jan-Feb, pp. 010402 1–, Jan. 2023.



04 Atomic and Molecular Physics