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

• URL: http://arxiv.org/abs/2603.29750v1
• Date: Tue, 31 Mar 2026 13:52:11 GMT
• Title: A criterion for an effective discretization of a continuous Schrödinger spectrum using a pseudostate basis
• Abstract summary: We consider a Hamiltonian $hat H$ with a (partially) continuous spectrum and examine the zero-overlap condition.<n>For each projected pseudostate the condition implies the occurrence of zeros at all energies that correspond to the pseudo-continuum matrix eigenvalues.<n>The zero-overlap condition ensures that in, e.g., an ionizing laser-atom interaction process, transition probabilities obtained from the projection of a time-propagated pseudostate-expanded function onto eigenstates of $hat H$ are stable.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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