Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials

Abstract

We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.

Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential

“…Next, let us show that the last two cases can be discarded. To this end, we first assume that the root in (25) and (26) is real-valued. This implies µ = 0, such that the energy E completely disappears from the condition.…”

“…As a consequence, no stationary energies can be determined. If we assume that the root in ( 25) is imaginary, then (25) results in n = −1, which is not a valid assignment due to the restriction that n must be a nonnegative integer. Finally, if the root in (26) takes imaginary values, we obtain n = 0 and |µ(E)| = k y / √ 2.…”

“…Energy-dependent potentials appear in a variety of applications, such as hydrodynamics [18], confined quantum systems [19] [27], or multi-nucleon systems [20]. Theoretical applications include the generation of nonrelativistic models with energy-dependent potentials by means of the supersymmetry formalism [26] and through point transformations [7] [25]. In the nonrelativistic context, the presence of energy-dependent potentials require a modification of the underlying quantum theory, principally affecting orthogonality relation and norm [6].…”

Dirac equation in the curved spacetime and generalized uncertainty principle: A fundamental quantum mechanical approach with energy-dependent potentials

“…The energy dependent potentials find place in various branches of physics such as relativistic quantum mechanics [17], semiconductors [18], high energy physics [19]. Among the investigation of mathematical complexities, one can be guided to the extension of the energy dependent potentials to the exceptional orthogonal polynomials and PT symmetry [20], [21], and low dimensional Dirac Hamiltonians [22], [23].…”