Axel Schulze-Halberg

Special function solutions of a spectral problem for a nonlinear quantum oscillator

We construct exact solutions of a spectral problem involving the Schrodinger equation for a nonlinear, one-parameter oscillator potential. In contrast to a previous analysis of the problem (Carinena et al 2007 Ann. Phys. 322 434–59), where solutions were given through a Rodrigues-type formula, our approach leads to closed-form representations of the solutions in terms of special functions, not containing any derivative operators. We show normalizability and orthogonality of our solutions, as well as correct reduction of the problem to the harmonic oscillator model, if the parameter in the potential gets close to zero.

DARBOUX TRANSFORMATIONS FOR TIME-DEPENDENT SCHRÖDINGER EQUATIONS WITH EFFECTIVE MASS

The formalism of Darboux transformations is established for time-dependent Schrodinger equations with an effective (position-dependent) mass. Explicit formulas are obtained for the transformed wave function and the difference between the original and the transformed potential. It is shown that for a noneffective mass our Darboux transformation reduces correctly to the well-known Darboux transformation.

EFFECTIVE MASS HAMILTONIANS WITH LINEAR TERMS IN THE MOMENTUM: DARBOUX TRANSFORMATIONS AND FORM-PRESERVING TRANSFORMATIONS

We define form-preserving transformations and Darboux transformations for time-dependent, effective mass Hamiltonians with additional linear terms. We give reality conditions for both transformations, guaranteeing the transformed potential to be real-valued. We further show that our form-preserving transformation preserves normalizability of the Schrodinger wave function. Our results generalize all former results on form-preserving transformations and Darboux transformations for the time-dependent Schrodinger equation. This paper is a sequel of Refs. 16–18.

On integral and differential representations of Jordan chains and the confluent supersymmetry algorithm

We construct a relationship between integral and differential representation of second-order Jordan chains. Conditions to obtain regular potentials through the confluent supersymmetry algorithm when working with the differential representation are obtained using this relationship. Furthermore, it is used to find normalization constants of wave functions of quantum systems that feature energy-dependent potentials. Additionally, this relationship is used to express certain integrals involving functions that are solution of Schrodinger equations through derivatives.

Explicit Darboux transformations of arbitrary order for generalized time-dependent Schrödinger equations

We construct Darboux transformations of arbitrary order for a generalized, linear, time-dependent Schrodinger equation, special cases of which correspond to time-dependent Hamiltonians coupled to a magnetic field, with position-dependent mass and with weighted energy. Our Darboux transformation reduces correctly to these known cases and also to new, generalized Schrodinger equations. Furthermore, fundamental properties of the conventional Darboux transformation are maintained, such as factorization of the nth order transformation into first-order transformations and existence of a reality condition for the transformed potentials.

Quantum systems with effective and time-dependent masses: form-preserving transformations and reality conditions

We study the time-dependent Schrödinger equation (TDSE) with an effective (position-dependent) mass, relevant in the context of transport phenomena in semiconductors. The most general form-preserving transformation between two TDSEs with different effective masses is derived. A condition guaranteeing the reality of the potential in the transformed TDSE is obtained. To ensure maximal generality, the mass in the TDSE is allowed to depend on time also.

The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials

We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results.

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

We construct a rational extension of a recently studied nonlinear quantum oscillator model. Our extended model is shown to retain exact solvability, admitting a discrete spectrum and corresponding closed-form solutions that are expressed through Jacobi-type Xm exceptional orthogonal polynomials.

An exactly solvable three-dimensional nonlinear quantum oscillator

Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the s-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.

Darboux transformations for a generalized Dirac equation in two dimensions

We construct explicit Darboux transformations for a generalized, two-dimensional Dirac equation. Our results complement and generalize former findings for Dirac equations in two and three spatial dimensions. We show that as a particular case, our Darboux transformations are applicable to the two-dimensional Dirac equation in cylindrical coordinates and give several examples.