Deformed shape invariance symmetry and potentials in curved space with two known eigenstates

Abstract

We consider two families of extensions of the oscillator in a d-dimensional constantcurvature space and analyze them in a deformed supersymmetric framework, wherein the starting oscillator is known to exhibit a deformed shape invariance property. We show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build potentials with known ground and first excited states. To extend such results to any members of the two families, we devise a general method wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W + (r) (and its accompanying function W − (r)).

Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere

“…Some years ago, infinite families of quasi-exactly solvable (QES) extensions of the oscillator potential on the sphere or in a hyperbolic space with known ground and first-excited states were constructed [1] in a deformed supersymmetric (DSUSY) framework [2], wherein the starting oscillator is known to exhibit a deformed shape invariance (DSI) property [3].…”

“…On taking into account the known equivalence between Schrödinger equations in a space of constant curvature and those using a position-dependent (effective) mass (PDM) [8], the power of the method developed in Ref. 1 to build infinite families of QES potentials with two known eigenstates was further demonstrated for several combinations of PDM and potential pairs [9,10].…”

On the classical and quantum dynamics of a class of nonpolynomial oscillators

“…where ω 0 and λ are parameters, has drawn considerable interest in the literature due to its unique simple harmonic oscillatory behavior [1,2]. The nonlinear system (1) and its generalizations are continuously studied for many aspects such as 3-dimensional [3] and d-dimensional generalizations [4,5] and rational extensions of the potentials [6]. Higgs geometrically interpretated the above nonlinear system which emerged while generalizing the Cartesian coordinates of Euclidean geometry.…”

Comment on ‘Two-dimensional position-dependent massive particles in the presence of magnetic fields’

“…Over the last few decades, both classical and quantum mechanical particles endowed with position-dependent mass (PDM) have attracted much research (see, e.g. [1][2][3][4][5][6][7][8] and references cited therein). The study of a classical PDM-nonlinear oscillator by Mathews and Lakshmanan [8] has sparked and inspired a large number of research in different fields of study.…”

“…(16) Of course, it looks the same as that in (1) for a classical particle but this presentation is the one to be used for a PDM quantum particle in a magnetic interaction, with P j (x, y) as the j th component of the PDM-momentum operator (7) and takes the differential form [6]…”