## Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators.(English)Zbl 1448.81345

Summary: We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra.

### MSC:

 81Q65 Alternative quantum mechanics (including hidden variables, etc.) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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### References:

 [1] Milburn, G. J.; Holmes, C. A., Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator, Phys. Rev. Lett., 56, 21, 2237 (1986) [2] Gabrielse, G.; Dehmelt, H.; Kells, W., Observation of a relativistic, bistable hysteresis in the cyclotron motion of a single electron, Phys. Rev. Lett., 54, 6, 537 (1985) [3] Seznec, R.; Zinn-Justin, J., Summation of divergent series by order dependent mappings: application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys., 20, 7, 1398-1408 (1979) · Zbl 0495.65002 [4] Andrianov, A. A., The large N expansion as a local perturbation theory, Ann. Phys., 140, 1, 82-100 (1982) [5] Graffi, S.; Grecchi, V., The Borel sum of the double-well perturbation series and the Zinn-Justin conjecture, Phys. Lett. B, 121, 6, 410-414 (1983) [6] Caliceti, E.; Grecchi, V.; Maioli, M., Double wells: perturbation series summable to the eigenvalues and directly computable approximations, Commun. Math. Phys., 113, 4, 625-648 (1988) · Zbl 0645.35071 [7] Buslaev, V.; Grecchi, V., Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A, Math. Gen., 26, 20, 5541 (1993) · Zbl 0817.47077 [8] Bender, C. M.; Boettcher, S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 80, 5243-5246 (1998) · Zbl 0947.81018 [9] Mostafazadeh, A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys., 7, 1191-1306 (2010) · Zbl 1208.81095 [10] Bender, C. M.; Dorey, P. E.; Dunning, C.; Fring, A.; Hook, D. W.; Jones, H. F.; Kuzhel, S.; Levai, G.; Tateo, R., PT Symmetry in Quantum and Classical Physics (2019), World Scientific: World Scientific Singapore [11] Jones, H. F.; Mateo, J., An equivalent Hermitian Hamiltonian for the non-Hermitian $$- x^4$$ potential, Phys. Rev. D, 73, Article 085002 pp. (2006) [12] Figueira de Morisson Faria, C.; Fring, A., Time evolution of non-Hermitian Hamiltonian systems, J. Phys. A, 39, 9269-9289 (2006) · Zbl 1095.81026 [13] Mostafazadeh, A., Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator, Phys. Lett. B, 650, 2, 208-212 (2007) · Zbl 1248.81049 [14] Znojil, M., Time-dependent version of crypto-Hermitian quantum theory, Phys. Rev. D, 78, 8, Article 085003 pp. (2008) [15] Bíla, H., Adiabatic time-dependent metrics in PT-symmetric quantum theories (2009), arXiv preprint [16] Gong, J.; Wang, Q.-H., Time-dependent PT-symmetric quantum mechanics, J. Phys. A, Math. Theor., 46, 48, Article 485302 pp. (2013) · Zbl 1280.81037 [17] Fring, A.; Moussa, M. H.Y., Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians, Phys. Rev. A, 93, 4, Article 042114 pp. (2016) [18] Fring, A.; Frith, T., Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians, Phys. Rev. A, 95 (2017), 010102(R) [19] Fring, A.; Frith, T., Mending the broken PT-regime via an explicit time-dependent Dyson map, Phys. Lett. A, 381, 2318 (2017) · Zbl 1377.81058 [20] Mostafazadeh, A., Energy observable for a quantum system with a dynamical Hilbert space and a global geometric extension of quantum theory, Phys. Rev. D, 98, 4, Article 046022 pp. (2018) [21] Fring, A.; Moussa, M. H.Y., Non-Hermitian Swanson model with a time-dependent metric, Phys. Rev. A, 94, 4, Article 042128 pp. (2016) [22] Fring, A.; Frith, T., Metric versus observable operator representation, higher spin models, Eur. Phys. J. Plus, 133, 57 (2018) [23] Khantoul, B.; Bounames, A.; Maamache, M., On the invariant method for the time-dependent non-Hermitian Hamiltonians, Eur. Phys. J. Plus, 132, 6, 258 (2017) [24] Maamache, M.; Djeghiour, O. K.; Mana, N.; Koussa, W., Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians, Eur. Phys. J. Plus, 132, 9, 383 (2017) [25] Cen, J.; Fring, A.; Frith, T., Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems, J. Phys. A, Math. Theor., 52, 11, Article 115302 pp. (2019) [26] Fring, A.; Frith, T., Solvable two-dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime, J. Phys. A, Math. Theor., 51, 26, Article 265301 pp. (2018) · Zbl 1396.81079 [27] Fring, A.; Frith, T., Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians, Phys. Lett. A, 383, 2-3, 158-163 (2019) · Zbl 1428.81097 [28] Fring, A.; Frith, T., Eternal life of entropy in non-Hermitian quantum systems, Phys. Rev. A, 100, 1, Article 010102 pp. (2019) [29] Fring, A.; Frith, T., Time-dependent metric for the two-dimensional, non-Hermitian coupled oscillator, Mod. Phys. Lett. A, 35, 08, Article 2050041 pp. (2020) [30] Koussa, W.; Maamache, M., Pseudo-invariant approach for a particle in a complex time-dependent linear potential, Int. J. Theor. Phys., 1-14 (2020) [31] Fring, A.; Tenney, R., Time-independent approximations for time-dependent optical potentials, Eur. Phys. J. Plus, 135, 2, 163 (2020)
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