Morse potential, symmetric Morse potential and bracketed bound-state energies

Abstract

For the needs of non-perturbative quantum theory an upgraded concept of solvability is proposed. In a broader methodical context the innovation involves Schrödinger equations which are piecewise analytic and piece-wise solvable in terms of special (in our illustrative example, Whittaker) functions. In a practical implementation of our symbolic-manipulation-based approach we work with a non-analyticity in the origin. A persuasive advantage is then found in the both-sidedness of our iterative localization of the energies.

Displaced Harmonic Oscillator V ∼ min [(x + d)2, (x − d)2] as a Benchmark Double-Well Quantum Model

“…In the methodical context, a strengthening and/or independent complement of our present arguments may be also found in Ref. [11] where we sampled the methodical as well as practical gains of the approach by introducing a spiked, centrally symmetrized 1-D potential of the Morse-oscillator type. In spite of its non-analyticity in the origin the model was still shown to exhibit several features of the more conventional complete exact solvability.…”

“…After one inserts potential (2) in Schrödinger Equation (11) it is possible to assume the existence of the even-parity and/or odd-parity QES bound-state solutions in the most general form of the standard normalizable polynomial ansatz such that, on the positive half-axis,…”

“…In the non-polynomial exact solvability (NES) context the non-QES exact spectra and bound states of the parity-symmetric models proved obtainable, for a few paritysymmetric potentials, via the x = 0 matching of their exact special-function left and right parts ψ (±) n (x) of the wave functions. This enabled one to assign the NES status to the symmetrized exponential potential V(x) ∼ exp |x| (with ψ (±) n (x) expressed in terms of Bessel functions [8,9]), to its asymptotically decreasing alternative [10] and/or to the symmetrized Morse potential V(x) (i.e., to the sum V(x) of the two exponentials for which ψ (±) n (x) appeared proportional to a confluent hypergeometric function [11,12]).…”

Symmetrized exponential oscillator

“…In other words, we believe that the conventional families of the exactly solvable one-dimensional potentials (with their typical list provided by review paper [15]) should be complemented by the symmetric functions V (x) = V (−x) which are non-analytic in the origin but still tractable via special functions (cf. also several other comments [3, 16,17,18,19,20,21] in this respect). Once more, let us remind the readers that our choice and study of potential (2) was motivated by the need of interpolation between the ubiquitous harmonic oscillator V (HO) (x) = ω 2 x 2 and its equally easily solvable square-well alternative…”

Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

“…It was recently suggested [20,21,22,23,24] that, apart from those abovementioned interaction models, the ES status should also be attributed to less common ones, for which the real line of coordinates is splitted into subintervals wherein the potential admits different definitions, while being continuous on the whole line, and the wavefunctions remain piece-wise proportional to special functions, while being matched, as well as their first derivatives, at the subinterval limit points.…”