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Title: Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation

Abstract

We present a numerical method, implemented in a Fortran code RESON, for computing resonance of the radial one-dimensional Schroedinger equation, for an underlying potential that decays sufficiently fast at infinity. The basic approach is to maximize the time-delay function [tau]([lambda]) as in the LeRoy program TDELAY. We present some theory that allows a preliminary bracketing of the resonance and various ways of reducing the total work. Together with automatic meshsize selection this leads to a method that has proved efficient, robust, and extremely trouble-free in numerical tests. The code makes use of Marletta's Sturm-Liouville solver, SLO2F, due to go into the NAG library. 24 refs., 4 figs., 3 tabs.

Authors:
 [1]
  1. (Royal Military College of Science, Swindon (United Kingdom))
Publication Date:
OSTI Identifier:
7158792
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; (United States); Journal Volume: 112:2
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; INELASTIC SCATTERING; COMPUTERIZED SIMULATION; SCHROEDINGER EQUATION; NUMERICAL SOLUTION; DIFFERENTIAL EQUATIONS; EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; SCATTERING; SIMULATION; WAVE EQUATIONS 661100* -- Classical & Quantum Mechanics-- (1992-); 990200 -- Mathematics & Computers

Citation Formats

Pryce, J.D. Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation. United States: N. p., 1994. Web. doi:10.1006/jcph.1994.1095.
Pryce, J.D. Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation. United States. doi:10.1006/jcph.1994.1095.
Pryce, J.D. 1994. "Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation". United States. doi:10.1006/jcph.1994.1095.
@article{osti_7158792,
title = {Efficient, reliable computation of resonances of the one-dimensional Schroedinger equation},
author = {Pryce, J.D.},
abstractNote = {We present a numerical method, implemented in a Fortran code RESON, for computing resonance of the radial one-dimensional Schroedinger equation, for an underlying potential that decays sufficiently fast at infinity. The basic approach is to maximize the time-delay function [tau]([lambda]) as in the LeRoy program TDELAY. We present some theory that allows a preliminary bracketing of the resonance and various ways of reducing the total work. Together with automatic meshsize selection this leads to a method that has proved efficient, robust, and extremely trouble-free in numerical tests. The code makes use of Marletta's Sturm-Liouville solver, SLO2F, due to go into the NAG library. 24 refs., 4 figs., 3 tabs.},
doi = {10.1006/jcph.1994.1095},
journal = {Journal of Computational Physics; (United States)},
number = ,
volume = 112:2,
place = {United States},
year = 1994,
month = 6
}
  • The new numerical approach to calculation of modified Mathieu functions is proposed. These functions play an important role in theories of electron scattering from (highly) polarizable atoms, like alkalis. The algorithms we developed show very high accuracy in a wide range of energy and polarizability, which are the two principal parameters of the problem. The numerical scheme does not lose the accuracy in the so called {open_quotes}unstable{close_quotes} regions, where the characteristic exponent of Mathieu functions becomes complex. This stability makes possible the analytical continuation of these methods in the complex plane of parameters.
  • No abstract prepared.
  • No abstract prepared.
  • A method for adiabatic excitation and control of multiphase (N-band) waves of the periodic nonlinear Schroedinger (NLS) equation is developed. The approach is based on capturing the system into successive resonances with external, small amplitude plane waves having slowly varying frequencies. The excitation proceeds from zero and develops in stages, as an (N+1)-band (N=0,1,2,...), growing amplitude wave is formed in the (N+1)th stage from an N-band solution excited in the preceding stage. The method is illustrated in simulations, where the excited multiphase waves are analyzed via the spectral approach of the inverse scattering transform method. The theory of excitation ofmore » 0- and 1-band NLS solutions by capture into resonances is developed on the basis of a weakly nonlinear version of Whitham's averaged variational principle. The phenomenon of thresholds on the driving amplitudes for capture into successive resonances and the stability of driven, phase-locked solutions in these cases are discussed.« less