Analytical solutions of the Fokker-Planck equation for generalized Morse and Hulthén potentials. (English) Zbl 1334.35350

Summary: In the present contribution we analytically calculate solutions of the transition probability of the Fokker-Planck equation (FPE) for both the generalized Morse potential and the Hulthén potential. The method is based on the formal analogy of the FPE with the Schrödinger equation using techniques from supersymmetric quantum mechanics.


35Q84 Fokker-Planck equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] Caughey, TK, Nonlinear theory of random vibrations, Adv. Appl. Mech., 11, 209-253, (1971)
[2] Oksendal, B.: Stochastic Diferential Equations, 5th edn. Springer, Berlin (1988)
[3] Reif, F.: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York (1985)
[4] Risken, H.: The FokkerPlanck Equation: Method of Solution and Applications. Springer, Berlin (1989) · Zbl 0665.60084
[5] Polotto, F; Araujo, MT; Drigo, FE, Solutions of the Fokker-Planck equation for a Morse isospectral potential, J. Phys. A, 43, 015207, (2010) · Zbl 1181.82051
[6] Araujo, MT; Drigo, FE, A general solution of the Fokker-Planck equation, J. Stat. Phys., 146, 610-619, (2012) · Zbl 1235.82051
[7] Filho, ED; Ricotta, RM, Supersymmetry, variational method and hulthen potential, Mod. Phys. Lett. A, 10, 1613-1618, (1995)
[8] Bender, CM; etal., Variational ansatz for PJ-symmetric quantum mechanics, Phys. Lett. A, 259, 224-231, (1999) · Zbl 0948.81536
[9] Bagchi, B; Quesne, C, Sl(2, C) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues, Phys. Lett. A, 273, 285-292, (2000) · Zbl 1050.81546
[10] Bagchi, B; Quesne, C, Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework, Phys. Lett. A, 300, 18-26, (2002) · Zbl 0997.81036
[11] Maghsoodi, E; Hassanabadi, H; Zarrinkamar, S, Exact solutions of the Dirac equation with Pöschl-Teller double-ring-shaped Coulomb potential via the Nikiforov-Uvarov method, Chin. Phys. B, 22, 030302, (2013)
[12] Buslaev, V; Grecchi, V, Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A, 36, 5541-5549, (1993) · Zbl 0817.47077
[13] Delabaere, E; Pham, F, Eigenvalues of complex Hamiltonians with PT-symmetry. I, Phys. Lett. A, 250, 25-28, (1998)
[14] Witten, E, Dynamical breaking of supersymmetry, Nucl. Phys. B, 185, 513-554, (1981) · Zbl 1258.81046
[15] Cooper, F; Khare, A; Sukhatme, U, Supersymmetry and quantum mechanics, Phys. Rep., 251, 267, (1995)
[16] Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry in Quantum Mechanics. World Scientic, Singapore (2001) · Zbl 0988.81001
[17] Ho, C-L; Sasaki, R, Quasi-exactly solvable Fokker-Planck equations, Ann. Phys., 323, 883-892, (2008) · Zbl 1214.82077
[18] Mesa, ADS; Quesne, C; Smirnov, YF, Generalized Morse potential-symmetry and satellite potentials, J. Phys. A, 31, 321-335, (1998) · Zbl 0956.81009
[19] Roose, T; Chapman, SJ; Maini, PK, Mathematical models of avascular cancer, SIAM Rev., 49, 179-208, (2007) · Zbl 1117.93011
[20] Anderson, ARA; Quaranta, V, Integrative mathematical oncology, Nat. Rev. Cancer, 8, 227-234, (2008)
[21] Preziosi, L.: Cancer Modeling and Simulation. Chapman & Hall, Boca Raton (2003) · Zbl 1039.92022
[22] Araujo, RP; McElwain, DLS, A history of the study of solid tumour growth: the contribution of mathematical modelling, Bull. Math. Biol., 66, 1039-1091, (2004) · Zbl 1334.92187
[23] Deng, ZH; Fan, YP, A potential function of diatomic molecules, Shandong Univ. J., 7, 162, (1957)
[24] Peyrard, M; Bishop, AR, Statistical mechanics of a nonlinear model for DNA denaturation, Phys. Rev. Lett., 62, 2755, (1989)
[25] Feizi, H; Shojael, MR; Rajabi, AA, Shape-invariance approach on the D-dimensional hulthen plus Coulomb potential for arbitrary l-state, Adv. Stud. Theor. Phys., 6, 477-484, (2012) · Zbl 1255.81168
[26] Aydogdu, O; Arda, A; Sever, R, Scattering and bound state solutions of the asymmetric hulthen potential, Phys. Scr., 84, 25004, (2011) · Zbl 1263.81147
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