Eigenfunctions and matrix elements for a class of eigenvalue problems with staggered ladder spectra

We present an alternate solution to an eigenvalue problem which arises in the study of the Fokker-Planck equation for generalized Ornstein-Uhlenbeck processes. We obtain the staggered ladder spectra found previously but in addition we obtain the normalized eigenfunctions in terms of associated Laguerre polynomials. The representation of the eigenfunctions in this form greatly simplifies the evaluation of matrix elements required in calculating ensemble averages and correlation coefficients for various observables. The solution to the eigenvalue problem is given in the generic and general cases

Explicit schemes for time propagating many-body wavefunctions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical solution of the time-dependent Schr\"odinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first order differential equations that are characterized by a high degree of stiffness. We analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. We consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness. For the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure

Multiresolution schemes for time-scaled propagation of wave packets

We present a detailed analysis of the time scaled coordinate approach and its implementation for solving the time-dependent Schr\"odinger equation describing the interaction of atoms or molecules with radiation pulses. We investigate and discuss the performance of multi-resolution schemes for the treatment of the squeezing around the origin of the bound part of the scaled wave packet. When the wave packet is expressed in terms of B-splines, we consider two different types of breakpoint sequences: an exponential sequence with a constant density and an initially uniform sequence with a density of points around the origin that increases with time. These two multi-resolution schemes are tested in the case of a one-dimensional gaussian potential and for atomic hydrogen. In the latter case, we also use Sturmian functions to describe the scaled wave packet and discuss a multi-resolution scheme which consists in working in a sturmian basis characterized by a set of non-linear parameters. Regarding the continuum part of the scaled wave packet, we show explicitly that, for large times, the group velocity of each ionized wave packet goes to zero while its dispersion is suppressed thereby explaining why, eventually, the scaled wave packet associated to the ejected electrons becomes stationary. Finally, we show that only the lowest scaled bound states can be removed from the total scaled wave packet once the interaction with the pulse has ceased

Reformulation of the strong-field approximation for light-matter interactions

We consider the interaction of hydrogen-like atoms with a strong laser field and show that the strong field approximation and all its variants may be grouped into a set of families of approximation schemes. This is done by introducing an ansatz describing the electron wave packet as the sum of the initial state wave function times a phase factor and a function which is the perturbative solution in the Coulomb potential of an inhomogeneous time-dependent Schr\"odinger equation. It is the phase factor that characterizes a given family. In each of these families, the velocity and length gauge version of the approximation scheme lead to the same results at each order in the Coulomb potential. By contrast, irrespective of the gauge, approximation schemes belonging to different families give different results. Furthermore, this new formulation of the strong field approximations allows us to gain deeper insight into the validity of the strong field approximation schemes. In particular, we address two important questions: the role of the Coulomb potential in the output channel and the convergence of the perturbative series in the Coulomb potential. In all the physical situations we consider here, our results are compared to those obtained by solving numerically the time-dependent Schr\"odinger equation.Comment: 19 pages, 9 figures, submitted for publicatio

Continuous stochastic Schrodinger equations and localization

The set of continuous norm-preserving stochastic Schrodinger equations associated with the Lindblad master equation is introduced. This set is used to describe the localization properties of the state vector toward eigenstates of the environment operator. Particular focus is placed on determining the stochastic equation which exhibits the highest rate of localization for wide open systems. An equation having such a property is proposed in the case of a single non-hermitian environment operator. This result is relevant to numerical simulations of quantum trajectories where localization properties are used to reduce the number of basis states needed to represent the system state, and thereby increase the speed of calculation.Comment: 18 pages in LaTeX + 6 figures (postscript), uses ioplppt.sty. To appear in J. Phys.

Sturmian bases for two-electron systems in hyperspherical coordinates

We give a detailed account of an $\it{ab}$ $\it{initio}$ spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron-electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb sturmian functions of half integer angular momentum. We show that, in the case of H$^-$, the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for $S\rightarrow P$ transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method.Comment: 28 pages, 4 figure