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

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.

• The paper demonstrates that a one-dimensional image space of QHP guarantees the zero-overlap condition for pseudostate bases.
• It employs Feshbach projection and residual function formalism to rigorously validate the approach for free-particle and Coulomb systems.
• The results improve asymptotic stability in numerical simulations for photoionization, scattering, and laser-matter interactions.

A Criterion for Effective Discretization of the Continuous Schrödinger Spectrum via Pseudostate Bases

Overview

The paper addresses the discretization of Hermitian operators with partially continuous spectra, focusing specifically on the foundational properties of pseudostate bases obtained by diagonalizing the Hamiltonian $\hat{H}$ in a finite $L^2$ basis. The core contribution is a rigorous criterion—framed within the Feshbach projector formalism—for achieving the so-called zero-overlap condition: for pseudostates projected onto the continuum, the overlap vanishes at all discretized continuum eigenvalues except the eigenvalue corresponding to the given pseudostate. This property ensures asymptotic stability in transition probability calculations for ionization and scattering processes.

Technical Summary of the Zero-Overlap Condition

Feshbach Projection Formalism

The analysis introduces orthogonal projectors $\hat{P}$ (onto the finite pseudostate basis) and $\hat{Q} = \hat{1} - \hat{P}$. The Schrödinger equation is split into coupled projected equations. Diagonalizing $\hat{H}$ in the subspace $\mathcal{P}$ yields pseudostates and associated discrete pseudo-continuum eigenvalues $\{\varepsilon_\ell\}$.

A sufficient condition for the zero-overlap property is identified: the image space of the operator $\hat{Q}\hat{H}\hat{P}$ must be one-dimensional. This implies that all residual couplings between the pseudostate and true continuum states are captured by a single residual function. At all the pseudo-continuum energies except for the relevant eigenvalue, the projections of other pseudostates onto continuum eigenstates vanish exactly.

Residual Function Formalism

For each projected pseudostate, a residual function is constructed in the continuum representation. The zeros of this function coincide with all pseudo-continuum eigenvalues except the one associated with the specific pseudostate. This forms the mathematical basis for the observed "clumping" where pseudostates exhibit dominant contributions around their associated energies and negligible (zero-overlap) at other discretized energies.

Generalization and Rigorous Proofs

The criterion is shown to be satisfied for:

• The 1D free-particle Hamiltonian with a harmonic oscillator basis: Only the highest occupied oscillator couples out of the finite basis subspace, making the image space of $\hat{Q}\hat{H}\hat{P}$ one-dimensional. The explicit connection between the residual function's zeros and the discrete energies is analytically demonstrated.
• The 3D Coulomb problem with Laguerre basis functions: Through explicit manipulation (cf. Appendix), it is proved that the action of $\hat{Q}\hat{H}\hat{P}$ yields a one-dimensional image for each angular momentum $L^2$0, thus rigorously establishing the zero-overlap property in this context.

Numerical and Analytical Results

The detailed analysis provides closed-form relationships and explicit calculation protocols for residual functions in both the free-particle and Coulomb scenarios. These are corroborated by numerical evaluations, confirming that:

• Each pseudostate eigenfunction exhibits exact zeros at the locations of the other pseudostate energies in the true continuum representation.
• The squared eigenfunctions of the pseudostate basis show pronounced localization ("clumping") near the corresponding discretized energy—supporting prior empirical observations.
• The criterion's failure in bases where the image space is multi-dimensional is explicitly demonstrated; in this case, the zero-overlap property does not hold, which is quantitatively verified via the non-vanishing of the residual norm.

Implications for Physical Calculations

Practical Significance

The zero-overlap condition ensures asymptotic stability in transition probability calculations derived from time-propagated wavefunctions in pseudostate bases. Specifically, it eliminates spurious asymptotic channel couplings, a long-standing challenge in computational treatments of photoionization and scattering. The condition is thus highly desirable for accurate extraction of observable quantities in collision and laser-matter interaction calculations.

Theoretical Implications

The main result frames the zero-overlap condition as a property of the basis and Hamiltonian, independent of the detailed structure of the $L^2$1 basis, provided the key criterion is met. This generalizes previous results that relied on specific orthogonality and closure properties of Laguerre or oscillator functions, extending applicability and providing a formal foundation for future pseudostate basis design.

Potential Extensions and Future Directions

The criterion suggests that basis engineering—for instance, by ensuring the image space of $L^2$2 remains one-dimensional—can be systematically pursued for other physical Hamiltonians, including those relevant for molecules or condensed matter. While the criterion is sufficient but not necessary, bases such as Gaussian or Slater-type orbitals may admit approximate satisfaction of the zero-overlap property, and numerical investigations are ongoing.

Implementations for more complex systems, many-body Hamiltonians, or time-dependent field interactions, as well as exploring the impact of imperfect (but approximately one-dimensional) image spaces, are natural areas for future research. Further, analytic techniques to control side maxima and the departures from ideal clumping in high-dimensional $L^2$3-spaces warrant investigation, particularly for high-precision differential measurements.

Conclusion

The paper provides a rigorous and generalizable criterion for ensuring exact zeros in the overlap of pseudostates with continuum eigenstates at all but their associated energies, through the dimensionality of the image space of $L^2$4. The result unifies observations in the Coulomb-Laguerre and oscillator representations, systematizing the understanding of pseudostate discretizations and offering a foundation for robust physical calculations in ionization, scattering, and laser-driven dynamics. The implications span both practical computational stability and theoretical simplification in the discretization of continuous spectra.

Reference:

"A criterion for an effective discretization of a continuous Schrödinger spectrum using a pseudostate basis" (2603.29750)