Equidistant spectra of anharmonic oscillators

Abstract

Some representative potentials of the anharmonic-oscillator type are constructed. Some corresponding spectra-shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrodinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral-shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrodinger problem and a certain type of nonclassical orthogonal polynomials is established.

Supersymmetric quantum mechanics and Painlevé equations

“…We will see in this section that the difficulties in the analysis grow with the order of the PHA: for zeroth and first order the systems become the harmonic and radial oscillators respectively [4,47,49,50]. On the other hand, for second and third order PHA, the determination of the potentials reduces to find solutions of Painlevé IV and V equations, respectively [4,51].…”

“…The algebraic structure generated by {H, L − m , L + m } provides information about the spectrum of H, Sp(H) [6,19,47]. In fact, let us consider the mth-dimensional solution space of the differential equation…”

“…Now, we propose a closedchain of three first-order SUSY transformations [3,5,6,[52][53][54][55] so that…”

Conditionally exactly solvable potentials and exceptional orthogonal polynomials

“…1 Subsequently, it has been shown that there are a number of exactly solvable potentials whose solutions are given in terms of these exceptional orthogonal polynomials ͑EOPs͒. 7 Here our objective is to re-examine some earlier results 8,9 related to conditionally exactly solvable potentials 10 in the context of EOPs, supersymmetry, and polynomial algebras. 4 The relation between EOPs, second order supersymmetry, 5 and shape invariance 6 has been investigated.…”

SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations

“…It is important to note that general systems ruled by second-and third-order polynomial Heisenberg algebras, described by one-dimensional Schrödinger Hamiltonians having third and fourth-order differential ladder operators, are linked to the Painlevé IV and V equations respectively [30,[33][34][35][36][37][38][39][40][41][42][43]. This connection was exploited in [28] to generate solutions to the Painlevé IV equation, using the SUSY partner Hamiltonians for the truncated oscillator, where it was used that the initial system is ruled by the Heisenberg-Weyl algebra.…”