Supersymmetry, shape invariance, and exactly solvable potentials

Abstract

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ‘‘shape-invariant’’ potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

Application of conditional shape invariance symmetry to obtain the eigen-spectrum of the mixed potentialV(r)=ar+br2+Bera1, Chakrabarti2, Das3 2017Physics Letters A221290View full textAdd to dashboardCite

“…The connection between solvable potentials and supersymmetric quantum mechanics (SUSY QM) is an established fact [1,2]. Gendenshtein showed that whenever the special symmetry known as 'shape-invariance symmetry' is satisfied by the supersymmteric partner potentials V 1 and V 2 , the entire energy spectrum including eigenfunctions of the Hamiltonian can be calculated by purely algebraic means [3].…”

Generation of a Complete Set of Additive Shape-Invariant Potentials from an Euler Equation

“…These special cases generate all known conventional additive shape-invariant superpotentials [3,11], as shown in Table I. Now that we have considered these special cases, we can systematically obtain all possible solutions.…”

“…If the parameters differ only by an additive constant a iþ1 ¼ a i þ @, the potentials are called ''additive'' or ''translational'' shapeinvariant. All exactly solvable potentials discovered thus far that are expressible in terms of known functions are additive shape-invariant [3,11]. Several groups found these potentials by imposing various Ansätz [10,[12][13][14].…”

Study of Deuteron with Modified Screen Coulomb (Yakawa) Potential

“…They consist of a long range part which describes empirically the observed repulsion with a few parameters. (Dutt, et al, 1988). There are several reviews on the meson exchange approach to the NN interaction.…”