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A criterion for an effective discretization of a continuous Schrödinger spectrum using a pseudostate basis

Tom Kirchner, Marko Horbatsch

Abstract

We consider a Hamiltonian $\hat H$ with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of $\hat H$ in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator $\hat Q \hat H \hat P$, where $\hat P$ is the projection operator onto the subspace spanned by the basis and $\hat Q = \hat 1 - \hat P$ its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of $ \hat H $ are asymptotically stable.

A criterion for an effective discretization of a continuous Schrödinger spectrum using a pseudostate basis

Abstract

We consider a Hamiltonian with a (partially) continuous spectrum and examine the zero-overlap condition which involves the projection onto exact continuum eigenstates of a set of pseudostates obtained from the diagonalization of in a finite basis of square-integrable functions. For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues, except for the eigenenergy associated with that pseudostate. This feature was observed for the Coulomb continuum represented in a Laguerre basis [M. McGovern et al., Phys. Rev. A 79, 042707 (2009)] and later explained using special properties of the Laguerre functions [I. B. Abdurakhmanov et al., J. Phys. B 44, 075204 (2011)]. We establish that a sufficient condition for the zero-overlap condition to occur is that the image space of the operator , where is the projection operator onto the subspace spanned by the basis and its complement, has dimension one. We show that the condition is met for the one-dimensional free-particle problem by a basis of harmonic oscillator eigenstates and for the Coulomb problem by a Laguerre basis, thus offering an alternative proof for the latter case. The zero-overlap condition ensures that in, e.g., an ionizing collision or laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded system wave function onto eigenstates of are asymptotically stable.

Paper Structure

This paper contains 5 sections, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Zero-overlap criterion
  3. The one-dimensional free-particle problem represented in a harmonic oscillator basis
  4. The Coulomb problem represented in a Laguerre basis
  5. Conclusions

Figures (3)

  • Figure 1: Arbitrarily scaled residual function (\ref{['eq:chiqk']}) (thin black curve) for $N=2$ and eigenfunctions (\ref{['eq:osci-ef1']}) (blue curve) and (\ref{['eq:osci-ef2']}) (red curve) for $\omega=0.2$ plotted in dependence of the wave number $\kappa$. The vertical bars locate the wave numbers $\kappa_{1,2}=\sqrt{2\varepsilon_{1,2}}$ that correspond to the matrix eigenvalues (\ref{['eq:secosc2']}).
  • Figure 2: Arbitrarily scaled residual function (\ref{['eq:chiqk']}) (thin black curve) for $N=6$ and squared eigenfunctions $\varphi_{\ell}^2(\kappa)$ ($\ell=1,\ldots, 6)$ (colored curves) for $\omega=0.2$ plotted in dependence of the wave number $\kappa$.
  • Figure 3: Arbitrarily scaled residual function (\ref{['eq:laguerre-res']}) (thin black curve) and squared eigenfunctions obtained from diagonalizing the radial Hamiltonian (\ref{['eq:hrad']}) in a Laguerre basis for $l=0$, $\lambda_0=2$, and $N_0=12$ (colored curves) plotted in dependence of the wave number $\kappa$. All (eight) eigenfunctions for positive energy eigenvalues are included, but only seven of them are clearly visible since the largest eigenvalue occurs at a $\kappa$-value outside of the interval displayed (see text for details).