Oscar Rosas-Ortiz

The supersymmetric modified Pöschl-Teller and delta well potentials

New supersymmetric (SUSY) partners of the modified Poschl-Teller and the Diracs delta well potentials are constructed in closed form. The resulting one-parametric potentials are shown to be interrelated by a limiting process. The range of values of the parameters for which these potentials are free of singularities is exactly determined. The construction of higher-order SUSY partner potentials is also investigated.

Position-dependent mass oscillators and coherent states

The solution of the Schrodinger equation for a position-dependent mass quantum system is studied in two ways. First, the interaction is found which must be applied to a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the Glauber states.

Coherent states for Hamiltonians generated by supersymmetry

Coherent states are derived for one-dimensional systems generated by supersymmetry from an initial Hamiltonian with a purely discrete spectrum for which the levels depend analytically on their subindex. It is shown that the algebra of the initial system is inherited by its SUSY partners in the subspace associated with the isospectral part or the spectrum. The technique is applied to the harmonic oscillator, infinite well and trigonometric Poschl–Teller potentials.

Gamow–Siegert functions and Darboux-deformed short range potentials

Abstract Darboux-deformations of short range one-dimensional potentials are constructed by means of Gamow–Siegert functions (resonance states). Results include both Hermitian and non-Hermitian short range potentials which are exactly solvable. As illustration, the method is applied to square wells and barriers for which the transmission coefficient is a superposition of Fock–Breit–Wigner distributions. Resonance levels are calculated in the long lifetime limit by means of analytical and numerical approaches. The new complex potentials behave as an optical device which both refracts and absorbs light waves.

Non-Hermitian SUSY hydrogen-like Hamiltonians with real spectra

It is shown that the radial part of the hydrogen Hamiltonian factorizes as the product of two not mutually adjoint first-order differential operators plus a complex constant � .T he 1-SUSY approach is used to construct non-Hermitian operators with hydrogen spectra. Other non-Hermitian Hamiltonians are shown to admit an extra ‘complex energy’ at � .N ew self-adjoint hydrogen-like Hamiltonians are also derived by using a 2-SUSY transformation with complex conjugate pairs � ,¯ � .

Nonlocal supersymmetric deformations of periodic potentials

Irreducible second-order Darboux transformations are applied to the periodic Schrodinger operators. It is shown that for the pairs of factorization energies inside the same forbidden band they can create new nonsingular potentials with periodicity defects and bound states embedded in the spectral gaps. The method is applied to the Lame and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.

The phenomenon of Darboux displacements

For a class of Schrodinger Hamiltonians the supersymmetry transformations can degenerate to simple coordinate displacements. We examine this phenomenon and show that it distinguishes the Weierstrass potentials including the one-soliton wells and periodic Lame functions. A supersymmetric sense of the addition formula for the Weierstrass functions is elucidated.

Confluent hypergeometric equations and related solvable potentials in quantum mechanics

The connection between the Schrodinger and confluent hypergeometric equations is discussed. It is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems. That description includes the linear and N-dimensional harmonic oscillators, as well as the Coulomb and Morse potentials.

DISTORTED HEISENBERG ALGEBRA AND COHERENT STATES FOR ISOSPECTRAL OSCILLATOR HAMILTONIANS

The dynamical algebra associated with a family of isospectral oscillator Hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates. It is shown that this representation becomes similar to that of the standard Heisenberg algebra, and it is dependent on a parameter omega >or=0. We call it the distorted Heisenberg algebra, where omega is the distortion parameter. The corresponding coherent states for an arbitrary omega are derived, and some particular examples are discussed in detail. A prescription to produce the squeezing, by adequately selecting the initial state of the system, is given.

Generalized coherent states for time-dependent and nonlinear Hamiltonian operators via complex Riccati equations

Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases, the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfils a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.