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 SIGMA, 2010, Volume 6, 098, 18 pages (Mi sigma556)

Multi-Well Potentials in Quantum Mechanics and Stochastic Processes

Victor P. Berezovoj, Glib I. Ivashkevych, Mikhail I. Konchatnij

A. I. Akhiezer Institute of Theoretical Physics, National Scientific Center "Kharkov Institute of Physics and Technology", 1 Akademicheskaya Str., Kharkov, Ukraine

Abstract: Using the formalism of extended $N=4$ supersymmetric quantum mechanics we consider the procedure of the construction of multi-well potentials. We demonstrate the form-invariance of Hamiltonians entering the supermultiplet, using the presented relation for integrals, which contain fundamental solutions. The possibility of partial $N=4$ supersymmetry breaking is determined. We also obtain exact forms of multi-well potentials, both symmetric and asymmetric, using the Hamiltonian of harmonic oscillator as initial. The modification of the shape of potentials due to variation of parameters is also discussed, as well as application of the obtained results to the study of tunneling processes. We consider the case of exact, as well as partially broken $N=4$ supersymmetry. The distinctive feature of obtained probability densities and potentials is a parametric freedom, which allows to substantially modify their shape. We obtain the expressions for probability densities under the generalization of the Ornstein–Uhlenbeck process.

Keywords: supersymmetry; solvability; partial breaking of $N=4$ supersymmetry; stochastic processes

DOI: https://doi.org/10.3842/SIGMA.2010.098

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ArXiv: 1006.5917
MSC: 81Q60
Received: October 6, 2010; in final form December 1, 2010; Published online December 18, 2010
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Citation: Victor P. Berezovoj, Glib I. Ivashkevych, Mikhail I. Konchatnij, “Multi-Well Potentials in Quantum Mechanics and Stochastic Processes”, SIGMA, 6 (2010), 098, 18 pp.

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Fellows J.M., Smith R.A., “A new two-parameter family of potentials with a tunable ground state”, Journal of Physics A-Mathematical and Theoretical, 44:33 (2011), 335302
2. Berezovoj V.P., Konchatnij M.I., “Dynamics of Localized States in N=4 Susy Qm”, Phys. Part. Nuclei, 43:5 (2012), 654–658
3. Berezovoj V.P., Konchatnij M.I., “Dynamics of Localized States in Extended Supersymmetric Quantum Mechanics with Multi-Well Potentials”, J. Phys. A-Math. Theor., 45:22 (2012), 225302
4. Berezovoj V.P., Konchatnij M.I., Nurmagambetov A.J., “Tunneling Dynamics in Exactly Solvable Models with Triple-Well Potentials”, J. Phys. A-Math. Theor., 46:6 (2013), 065302
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