Revisiting (Quasi-)Exactly Solvable Rational Extensions of the Morse Potential

Abstract

The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials V A,B,ext (x), obtained from a conventional Morse potential V A−1,B (x) by the addition of a bound state below the spectrum of the latter, is re-obtained. More importantly, the existence of another family of extended potentials, strictly isospectral to V A+1,B (x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of 'enlarged' shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones. Running head: Rational Extensions of Morse Potential

Group theoretic approach to rationally extended shape invariant potentials

“…There are other rationally extended potentials [34] for which the usual shape invariance condition is not valid and instead they satisfy an unfamiliar extended SI condition. So it will be interesting to know whether there are potential algebra that describe these systems.…”

“…[1] and for m = 1 the obtained potentials are the rationally extended translationally shape invariant (w.r.t potential parameters) potentials whose solutions are in terms of X 1 Laguerre or X 1 Jacobi polynomials [5,6]. Later on, the rational extension of the other exactly solvable potentials has also been considered, but the solutions of these potentials are not in the forms of EOPs, rather they are in the form of some new types of polynomials [34]. In these potentials the usual SI property is no more valid, rather they exhibit an unfamiliar extended SI property in which the partner potential is obtained by translating both the potential parameter A (as in the conventional case) and m, the degree of the polynomial arising in the denominator.…”

Time dependent potentials associated with exceptional orthogonal polynomials

“…The following boundary-value problem, representing a rationally extended Morse potential, was considered in Ref. 71:…”

Rational extension and Jacobi-type X m solutions of a quantum nonlinear oscillator

“…recall that n ≥ m. It is important to note that the conditions (19) guarantee that the denominator in (21) does not have any zeroes inside or at the endpoints of the interval (− π 2 , π 2 ), such that the solutions (21) are free of singularities, and as such can be normalized. The solution set (25) forms an orthogonal set with respect to the Hilbert space L 2 (−π/2, π/2), that is, for n, l ∈ N, n, l ≥ m,…”