We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ？-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ？.
References
[1]
Darboux, G. Le？ons sur la Théorie Général des Surfaces, 2nd ed.; Gauthier-Villars: Paris, France, 1912.
[2]
Schr？dinger, E. A method of determining quantum-mechanical eigenvalues and eigenfunctions. Proc. R. Ir. Acad. 1940, A46, 9–16.
[3]
Schr？dinger, E. Further studies on solving eigenvalue problems by factorization. Proc. R. Ir. Acad. 1941, A46, 183–206.
[4]
Schr？dinger, E. The factorization of the hypergeometric equation. Proc. R. Ir. Acad. 1941, A47, 53–54.
Witten, E. Dynamical breaking of supersymmetry. Nucl. Phys. 1981, B185, 513–554.
[7]
Solomonson, P.; Van Holten, J.W. Fermionic coordinates and supersymmetry in quantum mechanics. Nucl. Phys. 1982, B196, 509–531.
[8]
Cooper, F.; Freedman, B. Aspects of supersymmetric quantum mechanics. Ann. Phys. 1983, 146, 262–288.
[9]
Note the constant has been added to the usual harmonic oscillator potential to insure that the groundstate energy of the system remains at zero. This constant allows us to factorize the Hamiltonian as a product of operators and .
[10]
Cooper, F.; Khare, A.; Sukhatme, U. Supersymmetry in Quantum Mechanics; World Scientific: Singapore, 2001.
[11]
Gangopadhyaya, A.; Mallow, J.; Rasinariu, C. Supersymmetric Quantum Mechanics: An Introduction; World Scientific: Singapore, 2010.
[12]
Thus, is an eigenstate of with an eigenvalue .
[13]
.
[14]
Bender, C.M.; Boettcher, S. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett. 1998, 80, 5243–5246.
[15]
Bender, C.M.; Brody, D.C.; Jones, H.F. Complex extension of quantum mechanics. Phys. Rev. Lett. 2002, 89, 270401–1.
[16]
Bender, C.M.; Berry, M.V.; Mandilara, A. Generalized PT symmetry and real spectra. J. Phys. A 2002, 35, L467–L471.
[17]
Znojil, M. SI potentials with PT symmetry. J. Phys. A 2000, 33, L61–L62.
[18]
Levai, Z. Exact analytic study of the PT-symmetry-breaking mechanism. Czech. J. Phys. 2004, 54, 77–84.
[19]
Znojil, M. Matching method and exact solvability of discrete Pt-symmetric square wells. J. Phys. A 2006, 39, 10247–10261.
[20]
Quesne, C.; Bagchi, B.; Mallik, S.; Bila, H.; Jakubsky, V. PT supersymmetric partner of a short-range square well. Czech. J. Phys. 2005, 55, 1161–1166.
[21]
Bagchi, B.; Quesne, C.; Roychoudhury, R. Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT Symmetry. Pramana 2009, 73, 337–347.
[22]
Miller, W., Jr. Lie Theory and Special Functions (Mathematics in Science and Engineering); Academic Press: New York,NY,USA, 1968.
[23]
Gendenshtein, L.E. Derivation of exact spectra of the schrodinger equation by means of supersymmetry. JETP Lett. 1983, 38, 356–359.
Balantekin, A.; Candido Ribeiro, M.; Aleixo, A. Algebraic nature of shape-invariant and self-similar potentials. J. Phys. A 1999, 32, 2785–2790.
[33]
We assume that as , the supersymmetry remains unbroken.
[34]
Dutt, R.; Khare, A.; Sukhatme, U. Supersymmetry, shape invariance and exactly solvable potentials. Am. J. Phys. 1998, 56, 163.
[35]
Cooper, F.; Ginocchio, J.; Khare, A. Relationship between supersymmetry and solvable potentials. Phys. Rev. D 1987, 36, 2458.
[36]
In the last line we have used the fact that . This implies that for any analytical function , we have .
[37]
Veselov, A.P.; Shabat, A.B. Dressing chains and spectral theory of the Schr？dinger operator. Funct. Anal. Appl. 1993, 27, 81–96.
[38]
These constraints are: .
[39]
We have used .
[40]
Dutt, R.; Gangopadhyaya, A.; Rasinariu, C.; Sukhatme, U. Coordinate realizations of deformed Lie algebras with three generators. Phys. Rev. A 1999, 60, 3482–3486, doi:10.1103/PhysRevA.60.3482.
[41]
Rocek, M. Representation theory of the nonlinear SU (2) algebra. Phys. Lett. B 1991, 255, 554–557.
[42]
Adams, B.G.; Cizeka, J.; Paldus, J. Lie algebraic methods and their applications to simple quantum systems. Advances in Quantum Chemistry, 19th ed.; Academic Press: New York,NY,USA, 1987.
[43]
In some cases these are additive constants and subtracted away. If we provide a common floor to all potentials, demanding that their groundstate energies be zero, we will find that all known solvable potentials pick up a dependent term.
[44]
Shabat, A. The infinite-dimensional dressing dynamical system. Inverse Probl. 1992, 8, 303–308.
[45]
Gangopadhyaya, A.; Mallow, J.V. Generating shape invariant potentials. Int. J. Mod. Phys. A 2008, 23, 4959–4978.
[46]
Bougie, J.; Gangopadhyaya, A.; Mallow, J.V. Generation of a complete set of additive shape-invariant potentials from an euler equation. Phys. Rev. Lett. 2010, 210402–1.
[47]
Bougie, J.; Gangopadhyaya, A.; Mallow, J.V. Method for generating additive shape invariant potentials from an euler equation. J. Phys. A 2012, 44, 275307–1.
[48]
Normalizability of the groundstate requires that be greater than zero. Since an increase in decreases , there can only be a finite number of increases.
[49]
By substituting into Equation (50) we find that shape invariance requires that .
[50]
Quesne, C. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. J. Phys. A 2008, 41, 392001–1.
[51]
Quesne, C. Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics. Sigma 2009, 5, 084–1.
[52]
Odake, S.; Sasaki, R. Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the wilson and Askey-Wilson polynomials. Phys. Lett. B 2009, 682, 130–136.
[53]
Odake, S.; Sasaki, R. Another set of infinitely many exceptional (Xl) laguerre polynomials. Phys. Lett. B 2010, 684, 173–176.
[54]
Tanaka, T. N-fold supersymmetry and quasi-solvability associated with X-2-laguerre polynomials. J. Math. Phys. 2010, 51, 032101–1.
[55]
Sree Ranjani, S.; Panigrahi, P.; Khare, A.; Kapoor, A.; Gangopadhyaya, A. Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition. J. Phys. A 2012, 055210–1.
[56]
Shiv Chaitanya, K.; Sree Ranjani, S.; Panigrahi, P.; Radhakrishnan, R.; Srinivasan, V. Exceptional polynomials and SUSY quantum mechanics. Available online: http://arxiv.org/pdf/1110.3738.pdf (accessed on 2 August 2012).
