A criterion for an effective discretization of a continuous Schr\"odinger spectrum using a pseudostate basis

Marko Horbatsch; Tom Kirchner

REVIEW 2 major objections 4 minor 20 references

A criterion for an effective discretization of a continuous Schr\"odinger spectrum using a pseudostate basis

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a one-dimensional image space for the Feshbach coupling operator QHP is a sufficient condition for the zero-overlap condition, giving pseudostates whose projections onto true continuum states vanish at all other pseud

desk verdict Useful general criterion, but the Coulomb/Laguerre proof has an algebra error in Eq. (A.4) and the abstract overclaims a missing example. read the letter →

arxiv 2603.29750 v2 pith:NB33ODIL submitted 2026-03-31 physics.atom-ph quant-ph

Tom Kirchner , Marko Horbatsch This is my paper · ORCID

classification physics.atom-phquant-ph

keywords zero-overlapconditionpseudostatesFeshbachprojectioncontinuousspectrumLaguerrebasisL2discretizationionizationprobabilitiesresidualfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a general criterion for when a finite square-integrable basis, after diagonalization of a Hamiltonian with continuous spectrum, yields pseudostates that 'decouple' from the true continuum at each other's energies. The criterion is that the operator QHP, which couples the basis subspace to its complement, have a one-dimensional image space: then a single residual function encodes all pseudostate eigenvalues as its zeros, and the zero-overlap condition follows automatically. The authors verify the criterion for the 1D free particle in a harmonic oscillator basis and for the Coulomb problem in a Laguerre basis, providing an alternative proof of a phenomenon observed in earlier ionization calculations. If the criterion holds, transition probabilities obtained by projecting a time-propagated pseudostate wavefunction onto exact continuum eigenstates are asymptotically stable—an essential property for reliable ionization and scattering simulations.

What carries the argument

Feshbach projection operators P (onto the finite L2 basis) and Q = 1 − P, together with the coupling operator QHP and its image space. When that image is one-dimensional, all information about the coupling is contained in one residual function χ^Q(κ) = ⟨κ|QHP|φ_ℓ⟩, whose zeros locate the matrix eigenvalues and force the zero-overlap condition; the squared norm Λ(κ) offers a direct numerical test of whether the condition holds.

What would settle it

Construct or find a Hermitian Hamiltonian and a finite L2 basis for which the image of QHP is one-dimensional but two eigenvalues of PHP are exactly degenerate, then compute the overlaps ⟨φ_ℓ'|κ⟩ at the degenerate energy; if any is nonzero, the sufficiency claim as stated fails. Alternatively, numerically evaluate Λ(κ) from Eq. (14) for the harmonic oscillator basis with the ground state removed (a multi-dimensional image case) and confirm it has no zeros, which would support the necessity of the one-dimensional condition.

Watch

Extended reading notes

Core claim

For a Hermitian Hamiltonian with a (partially) continuous spectrum, if the image space of QHP is one-dimensional, then the pseudo-continuum eigenvalues of PHP coincide exactly with the zeros of a single residual function χ^Q(κ), and each pseudostate |φ_ℓ⟩ satisfies ⟨φ_ℓ'|κ⟩ = 0 at all other pseudostate energies ε_ℓ' ≠ ε_ℓ. This zero-overlap condition is sufficient for the asymptotic stability of projected transition probabilities in time-dependent calculations, and the paper demonstrates it explicitly for the harmonic-oscillator-basis free particle and the Laguerre-basis Coulomb problem.

Load-bearing premise

The sufficiency proof assumes that the pseudostate eigenvalues of PHP are non-degenerate; if two coincide, the decoupling equation can be satisfied without the zero-overlap condition, so the theorem as stated (for 'any Hermitian Hamiltonian') does not cover that case.

Editorial extensions

If this is right

Any basis satisfying the one-dimensional QHP image criterion automatically yields exact zeros at all other pseudostate energies, removing a source of spurious channel coupling in coupled-channel ionization calculations.
The criterion provides a practical diagnostic: compute Λ(κ) for a candidate basis and check for zeros; if found, the basis is 'effectively decoupled' from the continuum.
The Laguerre-basis Coulomb result is explained without invoking special properties of Laguerre functions, making the phenomenon more general than previously thought.
The free-particle harmonic-oscillator example shows the criterion is not restricted to Coulomb potentials, suggesting it may apply to other L2 bases and Hamiltonians.
Removing a basis state generically makes the QHP image multi-dimensional and destroys exact zeros, as demonstrated by the oscillator-minus-ground-state counterexample.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The criterion may explain why well-designed bases like Gaussian or Slater orbitals sometimes show near-zero-overlap behavior: if their QHP image is nearly one-dimensional, approximate stability follows, and Λ(κ) near zero could quantify the error.
The zero-overlap structure might be linked to a quadrature rule in energy space, where the zeros of the residual function define an effective grid for continuum integrals; this could inspire new basis construction strategies.
Since the Appendix notes the same residual structure for the free-particle Coulomb analog, the criterion may extend to any potential with similar radial polynomial form, e.g., a/r^2 + b/r with general coefficients.
The asymptotic-stability result from Ref. [15] likely carries over to other observables computed by projection onto continuum eigenstates, such as photoelectron spectra or autoionization widths.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The general criterion in Sec. II is the valuable core. It reduces the zero-overlap condition to one-dimensionality of the image of QHP, and it is derived cleanly from the Feshbach equations with no fitted parameters. The harmonic-oscillator free-particle example is nice and actually worked out. That part should survive.

What is new: the criterion generalizes the Laguerre-specific proof of Abdurakhmanov et al. to arbitrary L2 bases, and the HO example is genuinely new. The paper earns credit for isolating why Laguerre bases behave so well.

Now the soft spots. First, the abstract says a "minimal set of simple momentum-space wave functions" is shown to satisfy the condition, but that example never appears in the body. That is an overclaim and should be removed or the example added.

Second, and more serious: the Appendix derivation of the Laguerre residual contains an algebra error. Take l=0, k=1. Direct differentiation gives H ζ = N e^{-x/2}(λ²/2 − λ − λ²x/8). Equation (A.4) with the printed coefficient (k+l−2)/λ gives N e^{-x/2}(−λ/2 − λ²x/8), missing λ(λ−1)/2 e^{-x/2}. The correct coefficient is (k+l) − 2/λ, not (k+l−2)/λ. Since Eq. (A.4) is the linchpin for the claim that QHP has one-dimensional image for Laguerre bases, the proof as printed does not establish that result. The numerics in Fig. 3 suggest the conclusion still holds, and the error looks fixable, but as written the central Coulomb demonstration is unsupported.

Third, the theorem in Sec. II assumes non-degenerate PHP eigenvalues. The abstract says "any Hermitian Hamiltonian," which is too broad. If two pseudostate eigenvalues coincide, Eq. (8) can be satisfied without zero-overlap. This is a scope limitation, not a fatal flaw, but the statement should be qualified.

Bottom line: the general criterion is a solid, useful contribution and belongs in the literature once the Appendix is corrected, the abstract trimmed, and the non-degeneracy assumption stated. The paper deserves serious peer review, but I would recommend the referee ask for a corrected Appendix before acceptance.

Referee Report

2 major / 4 minor

Summary. The paper proposes a sufficient condition for the zero-overlap phenomenon observed for pseudostate discretizations of continuous spectra: if the image space of Q H P, with P the projector onto the finite L2 basis and Q=1-P, is one-dimensional, then each pseudostate's overlap with exact continuum eigenstates has zeros at all other pseudostate energies. This is derived from Feshbach projections in Sec. II, illustrated for the 1D free particle in a harmonic-oscillator basis in Sec. III, and applied to the Coulomb problem in a Laguerre basis in Sec. IV, with the algebraic reduction in an Appendix. The paper also connects the condition to asymptotic stability of time-dependent ionization probabilities via Ref. [15]. The abstract additionally promises an example with 'a minimal set of simple momentum-space wave functions' that does not appear in the body.

Significance. If the result holds, it provides a unified, parameter-free explanation of a property that was previously proved only for Laguerre bases by special-function identities. The Feshbach-based criterion is simple and potentially useful for assessing and designing basis sets for coupled-channel ionization calculations. The derivation in Sec. II is clean, self-contained, and does not rely on fitted parameters. However, the Coulomb demonstration rests on an algebraic reduction in the Appendix that contains an error as printed, so the paper cannot be accepted in its present form. The structural conclusion is likely salvageable after a corrected calculation, and the non-degeneracy caveat also needs explicit treatment.

major comments (2)

[Appendix, Eq. (A.4)] The printed coefficient in the second term of Eq. (A.4) is incorrect. For l=0, k=1, direct differentiation of ζ=N x e^{-x/2} gives H ζ = N e^{-x/2}(λ²/2 − λ − λ² x/8), whereas Eq. (A.4) with coefficient (k+l−2)/λ_l = −1/λ gives N e^{-x/2}(−λ/2 − λ² x/8). The correct coefficient is (k+l) − 2/λ_l, not (k+l−2)/λ_l. Since Eq. (A.4) is used to derive the residual (35) and the coefficient α_kl in Eq. (A.13), the proof that Q H P has one-dimensional image for the Laguerre basis is invalid as printed. The conclusion may survive after correction (the residual remains proportional to x_l^l L_N^{2l+1} e^{-x_l/2} with a modified coefficient), but the algebra must be redone and the revised α_kl stated explicitly.
[Sec. II, Eq. (8) and Abstract] The sufficiency argument explicitly assumes non-degeneracy of the PHP eigenvalues ('Assuming non-degeneracy...'), but this hypothesis is omitted from the abstract's blanket claim that the one-dimensionality of the Q H P image is a sufficient condition for the zero-overlap condition for any Hermitian Hamiltonian. If two pseudostate eigenvalues coincide, Eq. (8) can be satisfied at the degenerate energy without forcing the zero-overlap condition for all other eigenvectors. The paper should either add the non-degeneracy assumption to the statement of the criterion or provide a precise treatment of the degenerate case, e.g., by redefining the zero-overlap condition modulo degenerate pseudostate manifolds.

minor comments (4)

[Abstract and Sec. III] The abstract promises that the condition is shown for 'a minimal set of simple momentum-space wave functions,' but no such example appears anywhere in the body. Either provide this example or remove the phrase.
[Sec. V] Typo: 'ovservation' should be 'observation' in the first paragraph of the Conclusions.
[Sec. IV, Fig. 3 caption] The caption says eight eigenfunctions are included but only seven are visible; the text explains this, but the caption could be clearer that one eigenvalue lies outside the plotted κ range.
[Eq. (12)-(13)] The notation χ^Q(κ) is introduced as a single residual function, but the relation to the state-specific constants α_ℓ in Eq. (13) is not fully transparent; a sentence clarifying that Eq. (13) holds for each ℓ with a common χ^Q(κ) would help.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the one-dimensional QHP-image criterion is derived from Feshbach algebra, not defined by the zero-overlap condition; the only self-citation (Ref. 15) is motivational.

full rationale

The derivation is self-contained. Section II starts from the Feshbach coupled equations and defines the residual via ⟨κ|QHP|φℓ⟩ = (E(κ)-εℓ)φℓ(κ). Under a one-dimensional QHP image, a single residual determines all eigenvalue-matching zeros and forces the zero-overlap condition. This is an algebraic implication, not an input fit. The oscillator application uses the standard tridiagonal recurrence and verifies the N=2 eigenvalues by explicit determinant; the Laguerre application derives Eq. (35) from recurrence relations in the Appendix. No parameter is fitted to the zero-overlap condition, and no predicted quantity is equal by construction to a fit. The only self-citation, Ref. [15] by Tsui and Kirchner, is used to assert that zero-overlap implies asymptotic stability in time-dependent projections; this is practical motivation imported from prior work, not part of the proof of the sufficient condition. Even if one doubts the algebraic details (e.g., Eq. A.4), that would be a correctness flaw, not circular equivalence by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard quantum-mechanical projection formalism plus two explicit basis-scale parameters. No fitted constants or invented entities appear. The non-degeneracy assumption is the main unstated-in-abstract caveat.

free parameters (2)

omega (oscillator frequency)
Scale parameter of the harmonic-oscillator basis for the 1D free particle; arbitrary, the proof's conclusion is independent of its value.
lambda_l (Laguerre scale per angular momentum)
Scale parameters of the Laguerre basis in Eq. (34); chosen by hand (e.g., lambda0=2 in the example), not fitted; the residual-function analysis holds for any value.

assumptions (5)

domain assumption Standard Hilbert-space formalism: H is Hermitian; continuum eigenstates |kappa> are delta-normalized.
Used throughout Sec. II to define overlaps and energies.
standard math Feshbach projector algebra: P and Q are orthogonal, QP=0, and the coupled equations (2)-(3) follow from the eigenvalue equation.
Basis of the derivation; standard result.
domain assumption Non-degeneracy of the pseudostate eigenvalues epsilon_l of PHP.
Explicitly assumed in Sec. II to conclude that Eq. (8) forces matching and zero-overlap; degenerate cases are not treated.
standard math Laguerre polynomial recurrence and integral identities (A.1)-(A.3) and (A.9).
Used in the Appendix to reduce the residual function to a single component.
standard math Bound states are orthogonal to continuum eigenstates of the same Hermitian Hamiltonian.
Used implicitly to exclude kernel states of QHP from the zero-overlap argument.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A criterion for an effective discretization of a continuous Schr\"odinger spectrum using a pseudostate basis." pith.science (2026). https://pith.science/paper/NB33ODIL

@misc{pith2026260329750,
  author       = {Pith},
  title        = {Pith review of: A criterion for an effective discretization of a continuous Schr\"odinger spectrum using a pseudostate basis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NB33ODIL}},
  note         = {Machine review of arXiv:2603.29750}
}

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.

Figures

Figures reproduced from arXiv: 2603.29750 by the authors.

**Figure 2.** FIG. 2. Arbitrarily scaled residual function (20) (thin black curve) for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Arbitrarily scaled residual function (35) (thin black curve) and squared eigenfunctions [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references

[15]

Electron emission in antiproton-hydrogen interactions studied with the one-centre basis generator method,

J. J. Tsui and T. Kirchner, “Electron emission in antiproton-hydrogen interactions studied with the one-centre basis generator method,” (2026), arXiv:2603.18301 [physics.atom-ph]

arXiv 2026
[1]

E. J. Heller, T. N. Rescigno, and W. P. Reinhardt, Phys. Rev. A8, 2946 (1973)

1973
[2]

H. A. Yamani and W. P. Reinhardt, Phys. Rev. A11, 1144 (1975)

1975
[3]

J. F. Reading and A. L. Ford, Journal of Physics B: Atomic and Molecular Physics12, 1367 (1979)

1979
[4]

B. H. Bransden and A. T. Stelbovics, Journal of Physics B: Atomic and Molecular Physics17, 1877 (1984)

1984
[5]

Macías, F

A. Macías, F. Martín, A. Riera, and M. Yáañez, International Journal of Quantum Chemistry 33, 279 (1988)

1988
[6]

A. T. Stelbovics, Journal of Physics B: Atomic, Molecular and Optical Physics22, L159 (1989). 14

1989
[7]

I. Bray, I. B. Abdurakhmanov, J. J. Bailey, A. W. Bray, D. V. Fursa, A. S. Kadyrov, C. M. Rawlins, J. S. Savage, A. T. Stelbovics, and M. C. Zammit, Journal of Physics B: Atomic, Molecular and Optical Physics50, 202001 (2017)

2017

Show all 20 references

[8]

Semiclassical close-coupling approaches,

N. Sisourat and A. Dubois, “Semiclassical close-coupling approaches,” inIon-Atom Collisions: The Few-Body Problem in Dynamic Systems, edited by M. Schulz (De Gruyter, Berlin, 2019) pp. 157–178

2019
[9]

Electron–atom, electron–ion, and electron–molecule collisions,

K. Bartschat, J. Tennyson, and P. Burke, “Electron–atom, electron–ion, and electron–molecule collisions,” inSpringer Handbook of Atomic, Molecular, and Optical Physics, edited by G. W. F. Drake (Springer International Publishing, Cham, 2023) pp. 725–750

2023
[10]

Ion-atom and atom-atom collisions,

T. Kirchner, A. L. Ford, and J. F. Reading, “Ion-atom and atom-atom collisions,” inSpringer Handbook of Atomic, Molecular, and Optical Physics, edited by G. W. F. Drake (Springer International Publishing, Cham, 2023) pp. 785–794

2023
[11]

Uhlmann, T

M. Uhlmann, T. Kunert, and R. Schmidt, Phys. Rev. E72, 036704 (2005)

2005
[12]

K. A. Hamer, H. Gharibnejad, L. Argenti, and N. Douguet, Atoms13, 92 (2025)

2025
[13]

McGovern, D

M. McGovern, D. Assafrão, J. R. Mohallem, C. T. Whelan, and H. R. J. Walters, Phys. Rev. A79, 042707 (2009)

2009
[14]

I. B. Abdurakhmanov, A. S. Kadyrov, I. Bray, and A. T. Stelbovics, Journal of Physics B: Atomic, Molecular and Optical Physics44, 075204 (2011)

2011
[16]

A. I. Bondarev, I. I. Tupitsyn, I. A. Maltsev, Y. S. Kozhedub, and G. Plunien, Eur. Phys. J. D69, 110 (2015)

2015
[17]

Feshbach, Annals of Physics19, 287 (1962)

H. Feshbach, Annals of Physics19, 287 (1962)

1962
[18]

Cohen-Tannoudji, B

C. Cohen-Tannoudji, B. Diu, and F. Laloë,Quantum Mechanics, 2nd ed., Vol. 1 (Wiley-VCH, Berlin, 2019) Chap. V

2019
[19]

C. J. Joachain,Quantum Collision Theory, Vol. 1 (North Holland, Amsterdam, 1975) Chap. 6

1975
[20]

I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, corrected and enlarged ed. (Academic Press, Orlando, 1980). 15

1980

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [15]

Electron emission in antiproton-hydrogen interactions studied with the one-centre basis generator method,

J. J. Tsui and T. Kirchner, “Electron emission in antiproton-hydrogen interactions studied with the one-centre basis generator method,” (2026), arXiv:2603.18301 [physics.atom-ph]

arXiv 2026

[2] [1]

E. J. Heller, T. N. Rescigno, and W. P. Reinhardt, Phys. Rev. A8, 2946 (1973)

1973

[3] [2]

H. A. Yamani and W. P. Reinhardt, Phys. Rev. A11, 1144 (1975)

1975

[4] [3]

J. F. Reading and A. L. Ford, Journal of Physics B: Atomic and Molecular Physics12, 1367 (1979)

1979

[5] [4]

B. H. Bransden and A. T. Stelbovics, Journal of Physics B: Atomic and Molecular Physics17, 1877 (1984)

1984

[6] [5]

Macías, F

A. Macías, F. Martín, A. Riera, and M. Yáañez, International Journal of Quantum Chemistry 33, 279 (1988)

1988

[7] [6]

A. T. Stelbovics, Journal of Physics B: Atomic, Molecular and Optical Physics22, L159 (1989). 14

1989

[8] [7]

I. Bray, I. B. Abdurakhmanov, J. J. Bailey, A. W. Bray, D. V. Fursa, A. S. Kadyrov, C. M. Rawlins, J. S. Savage, A. T. Stelbovics, and M. C. Zammit, Journal of Physics B: Atomic, Molecular and Optical Physics50, 202001 (2017)

2017