We consider a Hamiltonian $\widehat{H}$ 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 $\widehat{H}$ 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 pseudocontinuum 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 $\widehat{Q}\widehat{H}\widehat{P}$, where $\widehat{P}$ is the projection operator onto the subspace spanned by the basis and $\widehat{Q}=\widehat{1}-\widehat{P}$ 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 $\widehat{H}$ are asymptotically stable.

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• Atomic & molecular collisions
• Electronic excitation & ionization