Authors : Gangadhar Behera; Pramoda Kumar Samal; Biswanath Rath

Volume/Issue : Volume 9 - 2024, Issue 10 - October

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Google Scholar : https://tinyurl.com/355vheja

Scribd : https://tinyurl.com/mrx7w9jd

DOI : https://doi.org/10.5281/zenodo.14621484

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Abstract : We study the convergence of the Morse- Feshbach nonlinear perturbation series (MFNPS) series to find out the energy levels of a PT symmetric complex cubic anharmonic oscillator. Perturbation series on energy has been calculated up to 15th order for the ground state and the first excited state. All orders of the MFNPS are found to be real and positive for this non-Hermitian but PT - symmetric Hamiltonian. The convergent energy spectra nicely match with the results of calculation of matrix diagonalization method. Some discussions on wave functions have been made using the nonlinear series.

Keywords : Morse-Feshbach; Non-Linear Perturbation Series; Anharmonic Os Cillator; Energy Levels.

References :

1. C. M. Bender and T.T.Wu, “Anharmonic Oscillator”, Phys. Rev, Vol. 184, 1969, pp 1231-1260
2. G. Halliday and P. Suryani, “Anhar monic oscillator: A new approach”, Phys. Rev. D., Vol 21, 1980, pp 1529-1537.
3. E. J. Weniger, Phys. Rev. Lett. Vol 77, 1966, pp 2862.
4. W. Janke and H. Klenert, “Convergent Strong-Coupling Expansions from Diver gent Weak-Coupling Perturbation The ory”, Phys. Rev. Lett, Vol 75, 1995, 2787 2791.
5. B. Rath, Phys. Soc. Jpn. “Construction of a Convergent Perturbation Theory: Case Study of the Anharmonic Oscillator Ground State Energy”, Vol 66, 1997, pp 3693-3695.
6. B. Rath, Phys. Soc. Jpn. “A New Ap proach on Wave Function and Energy Level Calculation Through Perturbation Theory”, Vol 67, 1998, 3044-3049.
7. C. Bender and S. Boettcher, “Real Spec tra in Non-Hermitian Hamiltonians Having PT Symmetry”, Phys. Rev. Lett, Vol 80, 1998, pp 5243-4246.
8. D. Feranchuk, L. I. Komarov, I. V. Nichipor, A. P. Ulyanenkov,“Operator Method in the Problem of Quantum An harmonic Oscillator”, Ann. Phys., Vol 238, 1995, pp 370-440.
9. O. D. Skoromnik1, and I. D. Feranchuk, “Analytic approximation for eigenvalues of a class of PT-symmetric Hamiltoni ans”, Physical review A, Vol 96, 2017, pp 052102.
10. B. Rath, “Real spectra in some nega tive potentials: linear and nonlinear one dimensional PT-invariant quantum sys tems.”, Eur. J. Phys. Plus, Vol 136, 2021, pp 493.
11. C.Tang and A. Frolov, “Eigenvalue and Eigenfunction for the PT-symmetric Po tential V = −(ix)N”, arXiv:1701.07180 v2.
12. P. M. Morse and Feshbach, Methods of Theoretical Physics, Part-II (Mc Graw Hill, New York), 1963.
13. B. Rath, “Case Study of the Conver gency of Nonlinear Perturbation Series: Morse–Feshbach Nonlinear Series”, Int. J. Mod.Phys A, Vol 14, 1999, pp 2103-2115.
14. P. Siegl and D. Krejcirik, “On the metric operator for the imaginary cubic oscilla tor”, Phys. Rev. D, Vol 86, 2012, 121702.