The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems

With the exception of the harmonic oscillator, quantum wave packets usually spread as time evolves. This is due to the non-linear character of the classical equations of motion which makes the various components of the wave packet evolve at various frequencies. We show here that, using the non-linear resonance between an internal frequency of a system and an external periodic driving, it is possible to overcome this spreading and build non-dispersive (or non-spreading) wave packets which are well localized and follow a classical periodic orbit without spreading. From the quantum mechanical point of view, the non-dispersive wave packets are time periodic eigenstates of the Floquet Hamiltonian, localized in the non-linear resonance island. We discuss the general mechanism which produces the non-dispersive wave packets, with emphasis on simple realization in the electronic motion of a Rydberg electron driven by a microwave field. We show the robustness of such wave packets for a model one-dimensional as well as for realistic three-dimensional atoms. We consider their essential properties such as the stability versus ionization, the characteristic energy spectrum and long lifetimes. The requirements for experiments aimed at observing such non-dispersive wave packets are also considered. The analysis is extended to situations in which the driving frequency is a multiple of the internal atomic frequency. Such a case allows us to discuss non-dispersive states composed of several, macroscopically separated wave packets communicating among themselves by tunneling. Similarly we briefly discuss other closely related phenomena in atomic and molecular physics as well as possible further extensions of the theory. (C) 2002 Elsevier Science B.V. All rights reserved

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“…The eigenvalues can then be calculated using an efficient diagonalization routine such as the Lanczos algorithm [115][116][117][118].…”

Section: Hamiltonian Basis Sets and Selection Rulesmentioning

confidence: 99%

Non-dispersive wave packets in periodically driven quantum systems

2002

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“…The basis B is then ordered in order to represent the Schrödinger equation using band matrices as narrow as possible. The eigenvalue problem is then solved using the Lanczos algorithm [23] which makes it possible to compute a few eigenvalues in the range of interest.…”

Section: Numerical Implementationmentioning

confidence: 99%

Quantum three-body Coulomb problem in two dimensions

et al. 2002

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We study the three-body Coulomb problem in two dimensions and show how to calculate very accurately its quantum properties. The use of a convenient set of coordinates makes it possible to write the Schrödinger equation only using annihilation and creation operators of four harmonic oscillators, coupled by various terms of degree up to twelve. We analyse in details the discrete symmetry properties of the eigenstates. The energy levels and eigenstates of the two-dimensional helium atom are obtained numerically, by expanding the Schrödinger equation on a convenient basis set, that gives sparse banded matrices, and thus opens up the way to accurate and efficient calculations. We give some very accurate values of the energy levels of the first bound Rydberg series. Using the complex coordinate method, we are also able to calculate energies and widths of doubly excited states, i.e. resonances above the first ionization threshold. For the two-dimensional H − ion, only one bound state is found.

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“…27,[32][33][34][35][36] Whether such an approach is more efficient ͑requires less Hamiltonian operations͒ depends on how many Hamiltonian operations are required to evaluate the action of f (Ĥ ) on a Lanczos vector. Because experience with respect to this issue is mixed, 32,36 in the present work we simply use Ĥ .…”

Section: Introductionmentioning

confidence: 99%

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

Kroes

Wiesenekker

Baerends

et al. 1996

The Journal of Chemical Physics

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We investigate the usefulness of a hybrid method for scattering with resonances. Wave packet propagation is used to obtain the time-dependent wave function ⌿(t) up to some time T at which direct scattering is over. Next, ⌿(t) is extrapolated beyond T employing resonance eigenvalues and eigenfunctions obtained in a Lanczos procedure, using ⌿(T) as starting vector to achieve faster convergence. The method is tested on one two-dimensional ͑2D͒ and one four-dimensional ͑4D͒ reactive scattering problem, affected by resonances of widths 0.1-5 meV. Compared to long time wave packet propagation, the hybrid method allows large reductions in the number of Hamiltonian operations N H required for obtaining converged reaction probabilities: A reduction factor of 24 was achieved for the 2D problem, and a factor of 6 for the 4D problem.

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The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems

Cited by 141 publications

References 9 publications

Non-dispersive wave packets in periodically driven quantum systems

Non-dispersive wave packets in periodically driven quantum systems

Quantum three-body Coulomb problem in two dimensions

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

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