Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 098, 18 pages      arXiv:1006.5917
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Multi-Well Potentials in Quantum Mechanics and Stochastic Processes

Victor P. Berezovoj, Glib I. Ivashkevych and Mikhail I. Konchatnij
A.I. Akhiezer Institute of Theoretical Physics, National Scientific Center ''Kharkov Institute of Physics and Technology'', 1 Akademicheskaya Str., Kharkov, Ukraine

Received October 06, 2010, in final form December 01, 2010; Published online December 18, 2010

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.

Key words: supersymmetry; solvability; partial breaking of N=4 supersymmetry; stochastic processes.

pdf (605 kb)   ps (1144 kb)   tex (1247 kb)


  1. Kramers H.A., Brownian motion in field of force and the diffusion model of chemical reactions, Physica 7 (1940), 284-304.
  2. Hanggi P., Talkner P., Berkovec M., Reaction-rate theory: fifty years after Kramers, Rev. Modern Phys. 62 (1990), 251-341.
  3. Gammaitoni L., Hanggi P., Jung P., Marchesoni F., Stochastic resonance, Rev. Modern Phys. 70 (1998), 22-287.
  4. Anishchenko V.S., Neiman A.B., Moss F., Shimansky-Geier L., Stochastic resonance: noise enhanced order, Phys. Usp. 42 (1999), 7-36.
  5. Pitaevskii L.P., Bose-Einstein condensation in magnetic traps. Introduction in a theory, Phys. Usp. 41 (1998), 569-580.
  6. Pitaevskii L.P., Bose-Einstein condensates in a laser field, Phys. Usp. 49 (2006), 333-351.
  7. Tye S.H.H., A new view of the cosmic landscape, hep-th/0611148.
  8. Chang L.L., Esaki L., Tsu R., Resonant tunneling in semiconductor double barriers, Appl. Phys. Lett. 24 (1974), 593-598.
  9. Risken H., The Fokker-Planck equation. Methods of solution and applications, 2nd ed., Springer Series in Synergetics, Vol. 18. Springer-Verlag, Berlin, 1989.
  10. van Kampen N.G., Stochastic processes in physics and chemistry, North-Holland Publishing Company, Amsterdam, 1992.
  11. Cooper F., Khare A., Sukhatme U., Supersymmetry in quantum mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
  12. Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  13. Hongler M.-O., Zheng W.M., Exact results for the diffusion in bistable potentials, J. Statist. Phys. 29 (1982), 317-327.
  14. Hongler M.-O., Zheng W.M., Exact results for the diffusion in a class of asymmetric bistable potentials, J. Math. Phys. 24 (1983), 336-340.
  15. Jauslin H.R., Exact propagator and eigenfunctions for multistable models with arbitrarily prescribed N lowest eigenvalues, J. Phys. A: Math. Gen. 21 (1988), 2337-2350.
  16. Zheng W.M., The Darboux transformation and solvable double-well potential models for Schrödinger equations, J. Math. Phys. 25 (1984), 88-90.
  17. Turbiner A.V., Double well potential: perturbation theory, tunneling, WKB (beyond instantons), Internat. J. Modern Phys. A 25 (2010), 647-658, arXiv:0907.4485.
  18. Jentschura U.D., Zinn-Justin J., Instantons in quantum mechanics and resurgent expansions, Phys. Lett. B 596 (2004), 138-144, hep-ph/0405279.
  19. Surzhykov A., Lubasch M., Zinn-Justin J., Jentschura U.D., Quantum dot potentials: Symanzik scaling, resurgent expansions, and quantum dynamics, Phys. Rev. B 74 (2006), 205317, 13 pages, cond-mat/0609027.
  20. Pashnev A.I., One-dimensional supersymmetrical quantum mechanics with N≥2, Teoret. and Math. Phys. 69 (1986), 1172-1175.
  21. Berezovoj V.P., Pashnev A.I., N=2 supersymmetric quantum mechanics and the inverse scattering problem, Teoret. and Math. Phys. 74 (1988), 264-268.
  22. Berezovoj V.P., Pashnev A.I., Extended N=2 supersymmetric quantum mechanics and isospectral Hamiltonians, Z. Phys. C 51 (1991), 525-529.
  23. Berezovoj V.P., Ivashkevych G.I., Konchatnij M.I., Exactly solvable diffusion models in the framework of the extended supersymmetric quantum mechanics, Phys. Lett. A 374 (2010), 1197-1200, arXiv:1006.5917.
  24. Gendenshtein L.E., Derivation of exact spectra of Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  25. Andrianov A.A., Ioffe M.V., Spiridonov V.P., Higher-derivative supersymmetry and the Witten index, Phys. Lett. A 174 (1993), 273-279, hep-th/9303005.
  26. Andrianov A.A., Ioffe M.V., Nishniadze D.N., Polynomial SUSY in quantum mechanics and second derivative Darboux transformation, Phys. Lett. A 201 (1995), 103-110, hep-th/9404120.
  27. Samsonov B.F., New features in supersymmetry breakdown in quantum mechanics, Modern Phys. Lett. A 11 (1996), 1563-1567, quant-ph/9611012.
  28. Plyuschay M., Hidden nonlinear supersymmetries in pure parabolic systems, Internat. J. Modern Phys. A 15 (2000), 3679-3698, hep-th/9903130.
  29. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.
  30. Witten E., Constraints on supersymmetry breaking, Nuclear Phys. B 202 (1982), 253-316.
  31. Faux M., Spector D., A BPS interpretation of shape invariance, J. Phys. A: Math. Gen. 37 (2004), 10397-10407, quant-ph/0401163.
  32. Ivanov E.A., Krivonos S.O., Pashnev A.I., Partial supersymmetry breaking in N=4 supersymmetric quantum mechanics, Classical Quantum Gravity 8 (1991), 19-39.
  33. Jeffreys H., Swirles B., Methods of mathematical physics, 3rd ed., Cambridge University Press, Cambridge, 1956.
  34. Luban M., Persey D.L., New Schrödinger equations for old: inequivalence of the Darboux and Abraham-Moses constructions, Phys. Rev. D 33 (1986), 431-436.
  35. Persey D.L., New families of isospectral Hamiltonians, Phys. Rev. D 33 (1986), 1048-1055.
  36. Mielnik B., Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984), 3387-3389.
  37. Agudov N.V., Noise delayed decay of unstable states, Phys. Rev. E 57 (1998), 2618-2625.
  38. Agudov N.V., Malakhov A.N., Decay of unstable equilibrium and nonequilibrium states with inverse probability current taken into account, Phys. Rev. E 60 (1999), 6333-6343.
  39. Berezhkovskii A.M., Talkner P., Emmerich J., Zitserman V.Yu., Thermally activated traversal of an energy barrier of arbitrary shape, J. Chem. Phys. 105 (1996), 10890-10895.
  40. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental function, Vol. 2, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953.
  41. Abraham P.B., Moses H.E., Changes in potentials due to changes in the point spectrum: anharmonic oscillators with exact solutions, Phys. Rev. A 22 (1980), 1333-1340.
  42. Chaudhury S., Cherayil B.J., Approximate first passage time distribution for barrier crossing in a double well under fractional Gaussian noise, J. Chem. Phys. 125 (2006), 114106, 8 pages.
  43. Mondescu R.P., Muthukumar M., Statistics of an ideal polymer in a multistable potential: exact solutions and instanton approximation, J. Chem. Phys. 110 (1999), 12240-12249.

Previous article   Next article   Contents of Volume 6 (2010)