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Supersymmetry,Shape Invariance and Exactly Solvable Noncentral Potentials  [PDF]
Avinash Khare,Rajat K. Bhaduri
Physics , 1993, DOI: 10.1119/1.17698
Abstract: Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss in detail the example of the potential $$V(r,\theta,\phi)={\omega^2\over 4}r^2 + {\delta\over r^2}+{C\over r^2 sin^2\theta}+{D\over r^2 cos^2\theta} + {F\over r^2 sin^2\theta sin^2 \alpha\phi} +{G\over r^2 sin^2\theta cos^2\alpha\phi}$$ with 7 parameters.Other algebraically solvable examples are also given.
Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry  [PDF]
C. Quesne
Physics , 2008, DOI: 10.1088/1751-8113/41/39/392001
Abstract: We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schr\"odinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.
Exactly Solvable Hydrogen-like Potentials and Factorization Method  [PDF]
J. Oscar Rosas-Ortiz
Mathematics , 1998, DOI: 10.1088/0305-4470/31/50/012
Abstract: A set of factorization energies is introduced, giving rise to a generalization of the Schr\"{o}dinger (or Infeld and Hull) factorization for the radial hydrogen-like Hamiltonian. An algebraic intertwining technique involving such factorization energies leads to derive $n$-parametric families of potentials in general almost-isospectral to the hydrogen-like radial Hamiltonians. The construction of SUSY partner Hamiltonians with ground state energies greater than the corresponding ground state energy of the initial Hamiltonian is also explicitly performed.
On Exactly Solvable Potentials  [PDF]
Darwin Chang,We-Fu Chang
Physics , 1995,
Abstract: We investigate two methods of obtaining exactly solvable potentials with analytic forms.
Exact supersymmetry in the relativistic hydrogen atom in general dimensions -- supercharge and the generalized Johnson-Lippmann operator  [PDF]
Hosho Katsura,Hideo Aoki
Mathematics , 2004, DOI: 10.1063/1.2173173
Abstract: A Dirac particle in general dimensions moving in a 1/r potential is shown to have an exact N = 2 supersymmetry, for which the two supercharge operators are obtained in terms of (a D-dimensional generalization of) the Johnson-Lippmann operator, an extension of the Runge-Lenz-Pauli vector that relativistically incorporates spin degrees of freedom. So the extra symmetry (S(2))in the quantum Kepler problem, which determines the degeneracy of the levels, is so robust as to accommodate the relativistic case in arbitrary dimensions.
Quasi-Exactly Solvable Potentials with Two Known Eigenstates  [PDF]
V. M. Tkachuk
Physics , 1998, DOI: 10.1016/S0375-9601(98)00455-1
Abstract: A new supersymmetry method for the generation of the quasi-exactly solvable (QES) potentials with two known eigenstates is proposed. Using this method we obtained new QES potentials for which we found in explicit form the energy levels and wave functions of the ground state and first excited state.
New hydrogen-like potentials  [PDF]
David J. Fernandez C
Mathematics , 2000, DOI: 10.1007/BF00400506
Abstract: Using the modified factorization method introduced by Mielnik, we construct a new class of radial potentials whose spectrum for l=0 coincides exactly with that of the hydrogen atom. A limiting case of our family coincides with the potentials previously derived by Abraham and Moses
Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials  [PDF]
C. -L. Ho
Mathematics , 2011, DOI: 10.1063/1.3671966
Abstract: We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum and Darboux-B\"acklund transformations
Supersymmetry and Singular Potentials  [PDF]
Ashok Das,Sergio A. Pernice
Physics , 1999, DOI: 10.1016/S0550-3213(99)00541-6
Abstract: The breaking of supersymmetry due to singular potentials in supersymmetric quantum mechanics is critically analyzed. It is shown that, when properly regularized, these potentials respect supersymmetry, even when the regularization parameter is removed.
Hydrogen supersymmetry: A new method in perturbation theory  [PDF]
E. A. Muljarov
Physics , 2012, DOI: 10.1002/1521-3951(200211)234:1<463
Abstract: The O(4) supersymmetry of the hydrogen atom is utilized to construct a complete basis using only the bound state wave functions. For a large class of perturbations, an expansion of the electron (exciton) wave function into such a complete set reduces the perturbed Schroedinger equation to a standard eigenvalue problem with an equidistant unperturbed spectrum. A high-order polarizability of the ground state of the hydrogen atom in a static electric field is calculated via Rayleigh-Schroedinger perturbation theory for illustration of the new method.
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