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arXiv:2603.29750

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

Tom Kirchner, Marko Horbatsch

physics.atom-phquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• Revised Jul 21, 2026 (v2)
v1 · Mar 31, 2026v2 · Jul 21, 2026
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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 by a minimal set of simple momentum-space wave functions, 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.

DOI
10.1103/sx1j-837h
Comments
3 figures

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We consider a Hamiltonian H^\hat HH^ 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 H^\hat HH^ 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 Q^H^P^\hat Q \hat H \hat PQ^​H^P^, where P^\hat PP^ is the projection operator onto the subspace spanned by the basis and Q^=1^−P^\hat Q = \hat 1 - \hat PQ^​=1^−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 by a minimal set of simple momentum-space wave functions, 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 H^\hat HH^ are asymptotically stable.
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