Numerical approximation of the Schr\"odinger equation with concentrated potential

Lehel Banjai; Mar\'ia L\'opez-Fern\'andez

REVIEW 3 major objections 4 minor 26 references

Numerical approximation of the Schr\"odinger equation with concentrated potential

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the convolution quadrature weights for the Schrödinger Green's function can be replaced by a fixed Gauss quadrature expansion, reducing memory from O(N) to O(n0 + log N) while keeping O(N log N) complexity.

desk verdict First fast-and-oblivious convolution quadrature for the Schrödinger Green's function, with a solid 1D analysis and a real complexity claim that needs correcting: the memory-saving regime costs O(N^{3/2} log N), not O(N log N). read the letter →

arxiv 1908.01292 v2 pith:N2NWPJUX submitted 2019-08-04 math.NA cs.NA

Lehel Banjai , María López-Fernández This is my paper · ORCID

classification math.NAcs.NA MSC 65R2065L0665M1565M38

keywords convolutionquadratureSchrödingerequationconcentratedpotentialfastandobliviousalgorithmGaussLaplacetransformpointinteractionsmemoryreduction

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 introduces a fast and oblivious convolution quadrature method for the Schrödinger equation with point-concentrated potentials, targeting the integral-equation reformulation of the problem. It claims that, once n ≥ n0, the convolution weights of the Schrödinger kernel can be replaced by a fixed Gauss quadrature expansion whose coefficients are updated by a simple recursion. This reduces the per-step cost to O(NQ) and the storage from O(N) to O(n0 + NQ), with NQ = O((log $ε^{{-1}}$ + $d^{2}$/(n0 h)) log(T/h)). If correct, long-time simulations of highly oscillatory or blow-up-prone solutions become feasible at far less memory cost than existing FFT-based or naive implementations.

What carries the argument

The machinery is the real inversion formula for the convolution quadrature weights (Theorem 8), which deforms the contour to the negative real axis and expresses ω_n(x) as an integral of G(λ,x) e_n(−hλ) over [0, ξ] plus vertical-line remainders. The analysis bounds the remainders (Proposition 9), truncates the integral, and approximates it on a geometrically graded partition using Gauss–Jacobi quadrature for the first interval and Gauss–Legendre quadrature for later intervals, with error estimates in Theorems 12 and 14. The fast summation step replaces each weight by a sum over NQ quadrature nodes and stores only the short direct part (n0 weights) plus the NQ running sums Q_{ℓ,k}.

What would settle it

For a fixed tolerance and decreasing h in D = 2, use the paper's parameter choices (e.g., B = 3, L0 = A1/T, ξ = A0/h) to compute the convolution weights for the 2D kernel and compare with standard FFT-based convolution quadrature weights. If the maximum error exceeds the tolerance or grows as h shrinks, the unproved D = 2 extension fails; alternatively, check whether the bound in (3.4) suffices to control the new singularity term.

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Extended reading notes

Core claim

The central discovery is the algorithm of Section 5. For n ≥ n0, the history term in the discrete convolution is rewritten as a fixed quadrature sum: the CQ weights ω_n(x) are first expressed through a real integral representation (Theorem 8), then approximated by Gauss–Jacobi quadrature on an initial interval and Gauss–Legendre quadrature on geometrically graded intervals. In the fast summation, each historical weight is replaced by contributions from NQ quadrature nodes, and the running sums Q_{ℓ,k} are updated by the recursion Q_{ℓ,k} = r(−h x_k) Q_{ℓ−1,k} + q(−h x_k) f_{ℓ−n0−1}. This yields the claimed complexity and memory bounds while preserving the accuracy of standard convolution quadrature.

Load-bearing premise

The quadrature and truncation error bounds are proved only for D = 1; the advertised D = 2 and D = 3 algorithms rely on an expectation, not a proof, that the same estimates hold after treating the kernel singularity at the origin.

Editorial extensions

If this is right

Long-time simulations of the nonlinear Schrödinger equation with point potentials become feasible at O(N log N) cost and O(n0 + log N) storage, allowing the study of quantum beating and finite-time blow-up.
The method's memory advantage grows with the number M of point potentials and with spatial dimension, since each convolution evaluation uses the same compressed history.
The recursion-based history term permits a variable-step implementation, which FFT-based convolution quadrature does not naturally allow.
Numerical experiments indicate the method matches standard convolution quadrature results for both linear beating and nonlinear suppression of beating over long time horizons.

Reading between the lines

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

The same real-inversion-plus-geometric-grading strategy may apply to other non-sectorial transfer operators with exponential growth, such as damped wave kernels, potentially simplifying the bookkeeping of earlier oblivious algorithms.
A rigorous extension to D = 2 and D = 3 would likely need a separate treatment of the kernel singularity at λ = 0; the paper's expressions for G(λ,x) show the 2D case involves a Bessel function J0, which is smoother than the 1D cosh, so the first-interval analysis may be improvable.
For problems with many point potentials, the O(n0 + NQ) storage becomes the dominant advantage over FFT-based methods, making the method most impactful in multi-point or higher-dimensional settings.
The finite-precision cancellation noted in Remark 17 could be tested by comparing the n0 = O(h^{-1/2}) and n0 = O(h^{-1}) parameter choices on ill-conditioned data; the paper leaves this trade-off as a user parameter.

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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 paper is worth reading and worth refereeing, but the headline complexity claim is internally contradicted by the paper's own formulas. What is genuinely new: this is the first application of fast-and-oblivious convolution quadrature to the Schrödinger Green's kernel. The authors derive a real integral representation of the CQ weights, a truncation bound, and a geometric-interval Gauss quadrature compression that avoids the mosaic bookkeeping of earlier oblivious methods. The 1D quadrature analysis is detailed and honest, the experiments match the intended benchmarks, and the published MATLAB code makes the implementation reproducible. Reusing their own prior framework is not an issue here; the Schrödinger application and the new integral representation are distinct enough, and the parameters are chosen from error bounds rather than fitted to solutions.

The soft spots are real, but not hidden. First, the complexity summary is wrong. The abstract and conclusions promise O(N log N) time and O(n0 + log N) memory. Section 5, however, derives total cost O(n0 log^2 n0 + N NQ) with NQ = O((log epsilon^{-1} + d^2/(n0 h)) log(T/h)). With the advertised memory-saving choice n0 = O(h^{-1/2}) and fixed d > 0, the d^2/(n0 h) term is unbounded, giving NQ = O(h^{-1/2} log h^{-1}) and total cost O(h^{-3/2} log h^{-1}) = O(N^{3/2} log N). So the paper's own asymptotic analysis contradicts its abstract. This is not a subtle disagreement with external consensus; it is an internal inconsistency. It can be fixed by revising the claimed complexity to match the actual parameter regime, or by explicitly choosing n0 = O(h^{-1}) and stating the memory trade-off. Second, all the rigorous quadrature theorems are proven only for D = 1. The introduction advertises a family of algorithms for D = 1, 2, 3, but Section 4 explicitly restricts the analysis and says similar estimates are "expected" in higher dimensions. That is a legitimate limitation, but it should be labeled more prominently. Third, Remark 17 notes the integrand can grow like exp(d^2/(n0 h)), so the memory-saving choice may suffer destructive cancellation in finite precision. The authors acknowledge this and push n0 larger when needed, but it remains a practical fragility.

Who is this for: numerical analysts working on convolution quadrature, Schrödinger point-interaction models, and fast convolution methods. It deserves a serious referee, and the code repository strengthens the case. My recommendation: send it to peer review, but require the authors to either correct the complexity claim or justify a parameter regime under which O(N log N) actually holds. As written, the central algorithm is solid in 1D, but the advertised scope and complexity need revision.

Referee Report

3 major / 4 minor

Summary. The paper develops fast and oblivious convolution quadrature (CQ) algorithms for time convolution with the free Schrödinger Green's function, targeting Schrödinger equations with point-concentrated potentials in dimensions D=1,2,3. Starting from the CQ weight representation (2.9), the authors derive a real contour integral representation for the weights (Theorem 8), truncate the vertical contour and bound the truncation error (Proposition 9, Corollary 10). For D=1 they split the remaining interval into an initial interval treated by Gauss-Jacobi quadrature (Theorem 12) and geometrically growing intervals treated by Gauss-Legendre quadrature (Theorem 14, Corollary 16). Section 5 combines the local direct convolution for the first n0 weights with a recursive history term evaluated by the quadrature, giving per-step cost O(NQ) and storage O(n0 + NQ), with NQ = O((log epsilon^{-1} + d^2/(n0 h)) log(T/h)). Numerical experiments in Section 6 validate the weight approximation and solve 1D linear and nonlinear models, including the quantum beating problem and a near-blow-up case. The paper claims O(N log N) complexity and O(n0 + log N) storage in the abstract and conclusions.

Significance. If the central derivation is correct, the 1D contribution is valuable: it provides a tolerance-driven parameter-selection strategy for the quadrature nodes, a recursive memory-efficient evaluation of the history term, and reproducible numerical experiments showing long-time and blow-up simulations. The quadrature and truncation analysis is detailed and does not appear to be fitted to benchmarks; the parameter choices are derived from the error bounds. However, the advertised complexity statement is internally inconsistent with Section 5's own estimates for the demonstrated regime n0 = O(h^{-1/2}): the global cost is O(N^{3/2} log N), not O(N log N), and the storage is not O(log N). Since the main selling point of the paper is efficiency relative to FFT-based CQ, this inconsistency affects the central claim. The restriction of all quadrature theorems to D=1 also means the advertised D=1,2,3 family is only supported in one dimension.

major comments (3)

[§5 (after Eq. (5.1)), §7] The complexity claims in the abstract and §7 are not supported by the estimates in §5. With n0 = O(h^{-1/2}) and fixed d > 0, the formula just after (5.1) gives NQ = O((log epsilon^{-1} + d^2/(n0 h)) log(T/h)) = O(h^{-1/2} log h^{-1}), so the total cost O(n0 log^2 n0 + N NQ) becomes O(h^{-3/2} log h^{-1}) = O(N^{3/2} log N). This is asymptotically worse than FFT-based CQ's O(N log N) and contradicts the 'O(N log N)' claim in §7. Likewise the storage is O(n0 + NQ) = O(h^{-1/2} log h^{-1}), not O(n0 + log N). The efficiency claims should be corrected to match what §5 actually proves; the memory-saving contribution can still be stated, but not as an O(N log N) method for d > 0.
[§4, page 8] All quadrature and truncation theorems (Corollary 10, Theorems 12 and 14, Corollary 16) are proved only for D = 1, and the text explicitly says 'we restrict ourselves to D = 1' and 'we expect similar estimates' for D = 2, 3. Since the abstract and introduction advertise algorithms for D = 1, 2, 3 and the singularity treatment is different in higher dimensions, the paper's scope claim is not established. The authors should either restrict the claims to one dimension or provide the analogous analysis, or at least numerical convergence evidence, for D = 2, 3.
[Remark 17, §5 (n0 paragraph)] The finite-precision stability of the memory-saving regime is not demonstrated. Remark 17 notes that the integrand can grow like exp(c d^2/(n0 h)), so with n0 = O(h^{-1/2}) the quadrature terms may suffer catastrophic cancellation; §5 acknowledges this and suggests n0 = O(h^{-1}) to remove the difficulty, but the complexity conclusions are drawn for n0 = O(h^{-1/2}). The paper should either provide a finite-precision analysis or an experimental study of cancellation, and state the complexity in the regime in which the algorithm is actually stable.

minor comments (4)

[§6.5] In the text before Figure 6, 'Fugures 6–8' should be 'Figures 6–8'.
[Theorem 14] In the statement of Theorem 14, the function denoted h_{n,j}(epsilon) is defined with eta_theta but the argument epsilon is never introduced; this notation should be cleaned up, presumably to h_{n,j}(eta_theta).
[Introduction] The term 'mosaic-free' is used without definition; since the mosaics of the earlier algorithms are the main point of comparison, a brief explanation would help readers.
[§6.5, Figure 5] Figure 5 would be more reproducible with a statement of the hardware and software environment and whether the timings include the parameter optimization.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: quadrature parameters are chosen from error bounds and tolerances, and the central contour/quadrature derivation is new and self-contained.

full rationale

Walking the derivation chain: Section 3 derives the real integral representation (Theorem 8) by deforming the contour in the Cauchy formula (2.9); Section 4 bounds the truncation and Gauss-Jacobi/Gauss-Legendre quadrature errors from classical estimates; Section 5 then chooses n0, L0, J, and Q from those bounds, stating 'with the choice ε0 = tol/2 and ε = tol/(2(J+1)) we have that the total error is bounded by the tolerance tol'. Nothing is fitted to reproduce benchmark solutions: the numerical experiments are genuine comparisons against standard CQ, FFT-based CQ, and the independent results of [13,21]. The self-citations to [6,7,24] supply the real-inversion idea and technical lemmas (e.g., Lemmas 2-4) that are independently published, parameter-free, and do not assume the Schrödinger result, so they do not make the argument circular. The main caveats are correctness and consistency issues rather than circularity: the sharp quadrature error estimates are proved only for D=1, with D=2,3 left as an expectation; Remark 17 warns about finite-precision cancellation; and the global complexity advertised as O(N log N) is not supported by the paper's own formula O(n0 log^2 n0 + N NQ) when n0 = O(h^{-1/2}), which gives O(h^{-3/2} log h^{-1}) = O(N^{3/2} log N). These are internal consistency and robustness concerns, not a reduction of the claimed prediction to its inputs.

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

The algorithm introduces no new physical entity, force, or conserved quantity. Its inputs are standard convolution quadrature machinery, the known Green's function, and standard special-function identities. The listed free parameters are algorithmic knobs set from error bounds and tolerances, not fitted constants reproducing the benchmark solutions. The main external reliance is on the authors' own earlier stability-function lemmas, which are independently published and not circular with the Schrödinger application.

free parameters (5)

A0 = 1 (truncation parameter via xi = A0/h)
Chooses the vertical truncation line in the contour integral. Larger A0 increases the number of direct weights n0 and the cost, smaller A0 increases the truncation error bound.
A1 = 2 (initial interval length via L0 = A1/T)
Sets the length of the first Gauss-Jacobi interval. The theory requires L0 proportional to T^{-1} to control the poles of the Runge-Kutta stability function.
B = 3 (geometric growth factor of quadrature intervals)
Defines the intervals L_j = (1+B) L_{j-1}. Smaller B gives more intervals and more quadrature nodes; larger B tightens the allowable Bernstein ellipse radius.
n0 = O(h^{-1/2}); examples 8, 15, 20, 27, 57 in experiments
Number of convolution weights computed directly before switching to the compressed history. It is the central memory-time tradeoff parameter and is chosen so the truncation error stays below tolerance.
NQ = 28, 48, 74, 112 in experiments; formula O((log eps^{-1} + d^2/(n0 h)) log(T/h))
Total number of quadrature nodes in the compressed history representation. Determined from the error bounds by solving a min-max problem numerically; not fitted to benchmark solutions.

assumptions (5)

domain assumption The Runge-Kutta method satisfies Assumption 1: A-stability, classical order p, stage order q, |r(iy)| < 1 for y != 0, invertible RK matrix, and stiff accuracy.
Used throughout to define the convolution quadrature weights and to prove the decay and growth properties of en(z). Satisfied by Radau IIA and Lobatto IIIC families, so it is a standard working assumption, not an ad hoc restriction.
standard math The Schrödinger Green's function k(t,x) has Laplace transform K(z,x) given by (1.3)-(1.4), analytic on C \ (-infinity, 0] with the stated algebraic bounds.
Taken from Economou and DLMF. This analyticity and growth behavior is what allows contour deformation in Lemma 6 and Theorem 8.
domain assumption Lemmas 2 and 3: there exist positive gamma(xi), nu, b, C_q such that |r(z)| <= e^{gamma(xi) Re z} in a strip and |r(z)| <= e^{nu Re z}, ||q(z)|| <= C_q for Re z <= b.
These stability-function estimates are proved in the authors' earlier papers [7] and [6] for RK methods satisfying Assumption 1. They are background results reused here and are central to the truncation and quadrature error bounds.
standard math Modified Bessel function asymptotics (3.2)-(3.4), especially |K0(z)| <= C e^{-Re z} |z|^{-1/2} for |arg z| < pi.
DLMF results needed to bound the 2D kernel and to justify the contour deformation in the proof of Theorem 8 and the subsequent quadrature estimates.
domain assumption Lemma 4: for Re z <= 0, r(z) = O(|z|^{-1}) and ||q(z)|| <= C_A |z|^{-1}, with C_A estimated numerically for Radau IIA and Lobatto IIIA methods.
Direct algebraic consequence of Assumption 1. It bounds the CQ weight vectors en(hz) and is used repeatedly in Corollary 10 and Theorems 12 and 14.

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Cite this review

Pith. "Pith review of Numerical approximation of the Schr\"odinger equation with concentrated potential." pith.science (2026). https://pith.science/paper/N2NWPJUX

@misc{pith2026190801292,
  author       = {Pith},
  title        = {Pith review of: Numerical approximation of the Schr\"odinger equation with concentrated potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2NWPJUX}},
  note         = {Machine review of arXiv:1908.01292}
}

read the original abstract

We present a family of algorithms for the numerical approximation of the Schr\"odinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms. These algorithms are special implementations of Lubich's Convolution Quadrature which allow, for certain applications in particular parabolic problems, to significantly reduce the computational cost and memory requirements. Recently it has been noticed that their use can be extended to some hyperbolic problems. Here we propose a new family of such efficient algorithms tailored to the features of the Green's function for Schr\"odinger equations. In this way, we are able to keep the computational cost and the storage requirements significantly below existing approaches. These features allow us to perform reliable numerical simulations for longer times even in cases where the solution becomes highly oscillatory or seems to develop finite time blow-up. We illustrate our new algorithm with several numerical experiments.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Adami, G

R. Adami, G. Dell’Antonio, R. Figari, and A. Teta. The Cau chy problem for the Schr¨ odinger equation in dimension three with concentrated nonlinearit y. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 20(3):477–500, 2003

work page 2003

[2] [2]

Adami, G

R. Adami, G. Dell’Antonio, R. Figari, and A. Teta. Blow-u p solutions for the Schr¨ odinger equation in dimension three with a concentrated nonlineari ty. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 21(1):121–137, 2004

work page 2004

[3] [3]

Adami and A

R. Adami and A. Teta. A class of nonlinear Schr¨ odinger eq uations with concentrated nonlin- earity. J. Funct. Anal. , 180(1):148–175, 2001. 24 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 50 100 150 200 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Figure 9 . On the left we compare the solution |q1(t)|2 for σ = 0. 9 and two choices of t...

work page 2001

[4] [4]

L. Banjai. Multistep and multistage convolution quadra ture for the wave equation: Algorithms and experiments. SIAM J. Sci. Comput. , 32(5):2964–2994, 2010

work page 2010

[5] [5]

Banjai and M

L. Banjai and M. L´ opez-Fern´ andez. Accompanying codes published at GitHub. https://github.com/lehelb/fastCQSchroedinger DOI:10. 5281/zenodo.3553729, 2019

work page 2019

[6] [6]

Banjai and M

L. Banjai and M. L´ opez-Fern´ andez. Eﬃcient high order a lgorithms for fractional integrals and fractional diﬀerential equations. Numer. Math. , 141(2):289–317, 2019

2019

[7] [7]

Banjai, M

L. Banjai, M. L´ opez-Fern´ andez, and A. Sch¨ adle. Fast and oblivious algorithms for dissipative and two-dimensional wave equations. SIAM J. Numer. Anal. , 55(2):621–639, 2017

2017

[8] [8]

Banjai and C

L. Banjai and C. Lubich. An error analysis of Runge-Kutta convolution quadrature. BIT, 51(3):483–496, 2011

work page 2011