A large class of bound-state solutions of the Schrödinger equation via Laplace transform of the confluent hypergeometric equation

From the generalized Morse potential to a unified treatment of the D-dimensional singular harmonic oscillator and singular Coulomb potentials

“…These results for the generalized Morse potential is in agreement with those ones obtained in Ref. [58] via Laplace transform method.…”

“…In a recent paper [58], it was shown that the Schrödinger equation for all those exactly solvable problems mentioned above can be reduced to the confluent hypergeometric equation in such a way that it can be solved via Laplace transform method with closed-form eigenfunctions expressed in terms of generalized Laguerre polynomials. Connections between the Morse and those other potentials have also been reported.…”

New solutions of the D-dimensional Klein–Gordon equation via mapping onto the nonrelativistic one-dimensional Morse potential

“…Bound-state solutions demand +∞ −∞ dx |ψ| 2 = 1 and exist only when the generalized Morse potential has a well structure (V 1 < 0 and V 2 > 0). The eigenenergies are then given by (see, e.g., [31], [38])…”

“…Bound states for systems modelled by those potentials are computed exactly in nonrelativistic quantum mechanics. In a recent paper [31], it was shown that the Schrödinger equation for all those exactly solvable problems mentioned above can be reduced to the confluent hypergeometric equation in such a way that it can be solved via Laplace transform method with closed-form eigenfunctions expressed in terms of generalized Laguerre polynomials. Connections between the Morse and those other potentials have also been reported.…”

Frustating use of the Laplace transform for the quantum states of a particle in a box

“…For some years now the 1/x [4], Morse [5], N -dimensional harmonic oscillator [6], pseudoharmonic and Mie-type [7], and double Dirac delta [8] potentials have been solved for the Laplace transform. The one-dimensional quantum harmonic oscillator problem has been revisited [9], and it has been shown that the bound-state solutions of the Schrödinger equation whose eigenfunctions are expressed in terms of particular solutions of the confluent hypergeometric equation can be obtained by using the Laplace transform of the confluent hypergeometric equation [10]. In a recent paper diffused in the literature [11], the authors claimed to solve the quantum problem of a particle in a infinite square well potential via Laplace transform.…”