pith. sign in

arxiv: 2605.03355 · v2 · submitted 2026-05-05 · 🧮 math.NA · cs.NA

Optimal error bounds on the exponential wave integrator for nonlinear Schr\"odinger equations with highly singular potential

Weizhu Bao , Chushan Wang , Yifei Wu This is my paper

Pith reviewed 2026-05-08 18:46 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords nonlinear Schrödinger equationexponential wave integratorerror estimatessingular potentialStrichartz estimatesnumerical analysisconvergence order
0
0 comments X

The pith

The exponential wave integrator achieves optimal first-order L2 convergence for nonlinear Schrödinger equations even with potentials as singular as L^p_loc for p > d/2.

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

This paper proves that the first-order exponential wave integrator for the nonlinear Schrödinger equation maintains optimal L2-norm convergence rates when the potential belongs to L^p_loc spaces with p greater than d/2, which is nearly the weakest regularity allowing the continuous equation to be well-posed. For p greater than 2 the scheme attains a full first-order rate, with a slight reduction to almost first order when p equals 2, and further reduced but predictable rates when the potential is even rougher. The analysis combines discrete space-time Strichartz estimates to control stability with normal-form transformations and frequency decompositions to obtain sharp error bounds without extra regularity loss. These results are new in three dimensions for L2 potentials and push the numerical theory to the exact threshold of the existence theory.

Core claim

We establish an optimal first-order L2-norm convergence for the EWI for NLSE with L^p_loc potentials where p>2, with order 1^- for p=2, and reduced orders of (1-alpha) for d=1,2 and (1-3/2 alpha) for d=3 when d/2 < p <2, where alpha = d(1/p - 1/2), using discrete Strichartz estimates and normal form transformations; this reaches the threshold regularity matching the well-posedness of the NLSE.

What carries the argument

Discrete space-time Lebesgue spaces together with discrete Strichartz estimates for stability, combined with normal form transformation and frequency decompositions for sharp error bounds.

If this is right

  • The scheme remains stable and attains the expected order without any smoothing or extra regularity imposed on the potential.
  • Optimal first-order L2 convergence is obtained for the first time in three dimensions when the potential is merely L2.
  • The error bounds match exactly the regularity threshold required for local well-posedness of the continuous NLSE.
  • The same techniques establish reduced but positive convergence orders for potentials down to p just above d/2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to higher-order exponential integrators or to other dispersive equations whose well-posedness is limited by singular coefficients.
  • Numerical experiments with rough or fractal potentials could be designed to confirm the predicted reduction in order as p approaches d/2.
  • This closes the gap between analysis and computation for quantum simulations involving highly irregular external potentials.

Load-bearing premise

The analysis assumes that discrete Strichartz estimates hold for the numerical scheme and that normal form transformations plus frequency decompositions yield optimal bounds without additional regularity loss.

What would settle it

A numerical test computing the L2 error of the EWI on the NLSE with a fixed L2 potential in three dimensions at successively halved time steps, showing the observed rate fails to approach 1, would disprove the first-order claim.

Figures

Figures reproduced from arXiv: 2605.03355 by Chushan Wang, Weizhu Bao, Yifei Wu.

Figure 6.1
Figure 6.1. Figure 6.1: Errors in L 2 - and H1 -norms of the EWI for the NLSE (1.1): (a) 1D case, and (b) 2D case view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Errors in L 2 - and H1 -norms of the EWI for the NLSE (1.1): (a) 1D case, and (b) 2D case convergence proved in Theorem 1.2. It remains unclear whether the error bound in Theorem 1.2 is optimal for highly singular potentials in 2D. On the other hand, the observed 0.65-order rate is still slower than the temporal convergence rate proved in Proposition 4.1, which is of 0.75 order. This indicates again that… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Errors in L 2 - and H1 -norms of the EWI for the NLSE (1.1): (a) L 2 − -potential, and (b) L 12 7 − -potential Acknowledgment This work was partially supported by the Ministry of Education of Singapore under its AcRF Tier 1 funding grant A-8003584-00-00 (W. Bao). This work was supported in part by the National Natural Science Foundation of China Grant No. 12494544 (Y. Wu). References [1] Y. Alama Bronsar… view at source ↗
read the original abstract

We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schr\"odinger equation (NLSE) with a highly singular potential in $\mathbb{R}^d$ with $1\leq d \leq 3$. Our results deal with singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $p>\frac{d}{2}$ and $p\geq 1$, which is (almost) the weakest regularity of the potential required by the well-posedness of the NLSE. First, for $L^p_\text{loc}$-potentials with $p>2$, we establish an optimal first-order $L^2$-norm convergence for the EWI, with the convergence order slightly reduced to $1^-$ when $p=2$. To the best of our knowledge, the optimal first-order convergence for the three-dimensional $L^2$-potential is for the first time in the literature. The optimality of such an error bound is two-fold: (i) the first-order $L^2$-norm convergence is optimal for the EWI (and its higher-order versions) under the given $L^2$-regularity assumption on the potential, and (ii) to achieve the first-order $L^2$-norm convergence for the EWI, such an assumption is optimally weak. For more singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $\frac{d}{2} < p < 2$ and $p\geq 1$, we prove that the $L^2$-norm convergence is (almost) of $(1-\alpha)$-order when $d=1,2$, and of $(1-\frac{3}{2}\alpha)$-order when $d=3$, where $\alpha:=d(1/p - 1/2)$ when $d =1,2,3$, $p>1$ and $\alpha:=\frac{1}{2}^+$ when $d=1$, $p=1$. Notably, this result pushes the error estimate to the threshold regularity of the potential that matches the threshold regularity for the well-posedness of the NLSE, which is also for the first time. Two main ingredients are adopted in the proof: (i) the use of discrete space-time Lebesgue spaces together with discrete Strichartz estimates to establish the stability of the numerical scheme, and (ii) the use of normal form transformation and frequency decompositions to obtain optimal error bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper establishes optimal error estimates for the first-order exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation with highly singular potentials V in L^p_loc(R^d) for 1 ≤ d ≤ 3 and p > d/2. For p > 2 it proves first-order L^2 convergence (reduced to order 1^- when p=2); for d/2 < p < 2 it obtains convergence of order (1-α) in d=1,2 and (1-3α/2) in d=3, where α = d(1/p - 1/2) (or α=1/2^+ when d=1, p=1). The proof rests on two ingredients: discrete space-time Lebesgue spaces together with discrete Strichartz estimates to obtain stability, and normal-form transformations combined with frequency decompositions to derive the error bounds. The results are claimed to reach the threshold regularity for well-posedness of the continuous NLSE and to be new for the 3D L^2-potential case.

Significance. If the stated rates hold, the work is significant: it pushes rigorous error analysis of a standard time integrator to the minimal regularity threshold required by the underlying PDE, which had not been achieved before for the 3D L^2-potential. The combination of discrete Strichartz estimates in space-time Lebesgue norms with normal-form error analysis supplies a reusable technical framework for other dispersive problems with low-regularity coefficients. The optimality statements are two-fold (scheme order and potential regularity) and are falsifiable by numerical experiments, strengthening the contribution.

minor comments (3)
  1. [Abstract] Abstract, paragraph 2: the definition of α is given for p>1 but the special case α=1/2^+ for d=1, p=1 is stated separately; a single compact formula or explicit table would improve readability.
  2. [stability analysis] The stability argument (ingredient (i)) invokes discrete Strichartz estimates without an explicit reference or short derivation showing that the constants remain uniform for V ∈ L^p_loc with p ↓ d/2. Adding a brief remark or citation to the relevant discrete Strichartz literature would remove any ambiguity about endpoint behavior.
  3. [error analysis] The frequency-decomposition step in the normal-form argument should state the precise regularity loss (if any) incurred when the potential multiplier is applied after the discrete propagator; this would make the passage from stability to the final (1-α) or (1-3α/2) rates fully transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment, including the recommendation for minor revision. We appreciate the recognition of the significance of reaching the well-posedness threshold for the potential regularity and the novelty in the 3D L^2 case. Below we address the report point by point.

read point-by-point responses

  1. Referee: The paper establishes optimal error estimates for the first-order exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation with highly singular potentials V in L^p_loc(R^d) for 1 ≤ d ≤ 3 and p > d/2. For p > 2 it proves first-order L^2 convergence (reduced to order 1^- when p=2); for d/2 < p < 2 it obtains convergence of order (1-α) in d=1,2 and (1-3α/2) in d=3, where α = d(1/p - 1/2) (or α=1/2^+ when d=1, p=1). The proof rests on two ingredients: discrete space-time Lebesgue spaces together with discrete Strichartz estimates to obtain stability, and normal-form transformations combined with frequency decompositions to derive the error bounds. The results are claimed to reach the threshold regularity for well-posedness of the continuous NLSE and to be new for the 3D L^2-potential case.

    Authors: We confirm that the stated error rates and proof ingredients accurately reflect the manuscript. The claims regarding reaching the well-posedness threshold and novelty for the 3D L^2-potential case are supported by the analysis in Sections 3 and 4, where we explicitly compare to the known well-posedness results for the continuous problem. revision: no

  2. Referee: If the stated rates hold, the work is significant: it pushes rigorous error analysis of a standard time integrator to the minimal regularity threshold required by the underlying PDE, which had not been achieved before for the 3D L^2-potential. The combination of discrete Strichartz estimates in space-time Lebesgue norms with normal-form error analysis supplies a reusable technical framework for other dispersive problems with low-regularity coefficients. The optimality statements are two-fold (scheme order and potential regularity) and are falsifiable by numerical experiments, strengthening the contribution.

    Authors: We agree with this evaluation of the significance. The two-fold optimality is discussed in the introduction and conclusion, and we have included a brief remark on potential numerical verification in the revised version to highlight falsifiability. revision: partial

Circularity Check

0 steps flagged

No circularity; error analysis uses external estimates and standard techniques

full rationale

The derivation proceeds by comparing the EWI scheme to the continuous NLSE via discrete Strichartz estimates for stability and normal-form/frequency-decomposition arguments for error control. These are applied to the Duhamel formulation and do not reduce the target convergence rate to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose content is merely renamed. The optimality statements match the claimed rates to the known well-posedness threshold for the continuous problem, which is an external benchmark rather than an internal tautology. No equation or step in the abstract or described proof chain exhibits the specific reduction required for a circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard PDE well-posedness results and functional-analytic estimates without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption NLSE is well-posed for potentials in L^p_loc(R^d) with p > d/2 and p >= 1
    Invoked as the minimal regularity baseline that the numerical error analysis matches.
  • standard math Discrete Strichartz estimates hold for the exponential wave integrator
    Used to prove stability of the scheme in discrete space-time norms.

pith-pipeline@v0.9.0 · 5790 in / 1452 out tokens · 73146 ms · 2026-05-08T18:46:15.030113+00:00 · methodology

Review history (3 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    Alama Bronsard, Y

    Y. Alama Bronsard, Y. Bruned, and K. Schratz , Low regularity integrators via decorated trees , 2022, arXiv:2202.01171

    work page arXiv 2022

  2. [2]

    P. W. Anderson , Absence of diffusion in certain random lattices , Phys. Rev., 109 (1958), pp. 1492--1505

    work page 1958

  3. [3]

    R. Bai, Y. Lian, and Y. Wu , Regularization for the S chr\"odinger equation with rough potential: high-dimensional case , 2025, arXiv:2510.25555

    work page arXiv 2025

  4. [4]

    R. Bai, Y. Lian, and Y. Wu , Regularization for the S chr\"odinger equation with rough potential: one-dimensional case , 2025, arXiv:2510.25540

    work page arXiv 2025

  5. [5]

    Bao and Y

    W. Bao and Y. Cai , Mathematical theory and numerical methods for B ose- E instein condensation , Kinet. Relat. Models, 6 (2013), pp. 1--135

    work page 2013

  6. [6]

    W. Bao, B. Lin, Y. Ma, and C. Wang , An extended F ourier pseudospectral method for the G ross- P itaevskii equation with low regularity potential , East Asian J. Appl. Math., 14 (2024), pp. 530--550

    work page 2024

  7. [7]

    W. Bao, Y. Ma, and C. Wang , Optimal error bounds on time-splitting methods for the nonlinear S chr\" o dinger equation with low regularity potential and nonlinearity , Math. Models Methods Appl. Sci., 34 (2024), pp. 803--844

    work page 2024

  8. [8]

    Bao and C

    W. Bao and C. Wang , Error estimates of the time-splitting methods for the nonlinear S chr\" o dinger equation with semi-smooth nonlinearity , Math. Comp., 93 (2024), pp. 1599--1631

    work page 2024

  9. [9]

    Bao and C

    W. Bao and C. Wang , An explicit and symmetric exponential wave integrator for the nonlinear S chr\" o dinger equation with low regularity potential and nonlinearity , SIAM J. Numer. Anal., 62 (2024), pp. 1901--1928

    work page 2024

  10. [10]

    Bao and C

    W. Bao and C. Wang , Optimal error bounds on the exponential wave integrator for the nonlinear S chr\" o dinger equation with low regularity potential and nonlinearity , SIAM J. Numer. Anal., 62 (2024), pp. 93--118

    work page 2024

  11. [11]

    Error estimates of an exponential wave integrator for the nonlinear Schr\"odinger equation with singular potential

    W. Bao and C. Wang , Error estimates of an exponential wave integrator for the nonlinear S chr\"odinger equation with singular potential , SIAM J. Numer. Anal., to appear (arXiv:2504.03346)

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Becker, N

    S. Becker, N. Galke, R. Salzmann, and L. van Luijk , Convergence rates for the Trotter splitting for unbounded operators , Found. Comput. Math., (2025), DOI:10.1007/s10208-025-09730-w

    work page doi:10.1007/s10208-025-09730-w 2025

  13. [13]

    Blanes, F

    S. Blanes, F. Casas, and A. Murua , Splitting methods for differential equations , Acta Numerica, 33 (2024), p. 1–161

    work page 2024

  14. [14]

    Burgarth, P

    D. Burgarth, P. Facchi, A. Hahn, M. Johnsson, and K. Yuasa , Strong error bounds for T rotter and S trang-splittings and their implications for quantum chemistry , Phys. Rev. Res., 6 (2024), p. 043155

    work page 2024

  15. [15]

    Carles , Time splitting and error estimates for nonlinear S chr\"odinger equations with a potential , Found

    R. Carles , Time splitting and error estimates for nonlinear S chr\"odinger equations with a potential , Found. Comput. Math., (2025), DOI:10.1007/s10208-025-09727-5

    work page doi:10.1007/s10208-025-09727-5 2025

  16. [16]

    Carles and C

    R. Carles and C. Su , Scattering and uniform in time error estimates for splitting method in NLS , Found. Comput. Math., 24 (2024), pp. 683--722

    work page 2024

  17. [17]

    K. M. Case , Singular potentials , Phys. Rev., 80 (1950), pp. 797--806

    work page 1950

  18. [18]

    Cazenave , Semilinear Schr\" o dinger Equations , vol

    T. Cazenave , Semilinear Schr\" o dinger Equations , vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, 2003

    work page 2003

  19. [19]

    Choi and Y

    W. Choi and Y. Koh , On the splitting method for the nonlinear S chr\" o dinger equation with initial data in H^1 , Discrete Contin. Dyn. Syst., 41 (2021), pp. 3837--3867

    work page 2021

  20. [20]

    Erd o s, B

    L. Erd o s, B. Schlein, and H.-T. Yau , Derivation of the cubic non-linear S chr\" o dinger equation from quantum dynamics of many-body systems , Invent. Math., 167 (2007), pp. 515--614

    work page 2007

  21. [21]

    D. Fang, X. Wu, and A. Soffer , On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials , 2025, preprint (arXiv:2507.22707)

    work page arXiv 2025

  22. [22]

    Fibich , The Nonlinear Schr\" o dinger Equation: Singular Solutions and Optical Collapse , vol

    G. Fibich , The Nonlinear Schr\" o dinger Equation: Singular Solutions and Optical Collapse , vol. 192 of Applied Mathematical Sciences, Springer, Cham, 2015

    work page 2015

  23. [23]

    W. M. Frank, D. J. Land, and R. M. Spector , Singular potentials , Rev. Mod. Phys., 43 (1971), pp. 36--98

    work page 1971

  24. [24]

    Gross and I

    C. Gross and I. Bloch , Quantum simulations with ultracold atoms in optical lattices , Science, 357 (2017), pp. 995--1001

    work page 2017

  25. [25]

    Henning and D

    P. Henning and D. Peterseim , Crank- N icolson G alerkin approximations to nonlinear S chr\" o dinger equations with rough potentials , Math. Models Methods Appl. Sci., 27 (2017), pp. 2147--2184

    work page 2017

  26. [26]

    Hochbruck and A

    M. Hochbruck and A. Ostermann , Exponential integrators , Acta Numer., 19 (2010), pp. 209--286

    work page 2010

  27. [27]

    L. I. Ignat , A splitting method for the nonlinear S chr\" o dinger equation , J. Differential Equations, 250 (2011), pp. 3022--3046

    work page 2011

  28. [28]

    Jahnke and C

    T. Jahnke and C. Lubich , Error bounds for exponential operator splittings , BIT, 40 (2000), pp. 735--744

    work page 2000

  29. [29]

    B. Lin, Y. Ma, and C. Wang , A L awson-time-splitting extended F ourier pseudospectral method for the G ross- P itaevskii equation with time-dependent low regularity potential , J. Comput. Phys., 512 (2024), p. 113133

    work page 2024

  30. [30]

    o dinger- P oisson and cubic nonlinear S chr\

    C. Lubich , On splitting methods for S chr\" o dinger- P oisson and cubic nonlinear S chr\" o dinger equations , Math. Comp., 77 (2008), pp. 2141--2153

    work page 2008

  31. [31]

    N. J. Mauser, Y. Wu, and X. Zhao , The cubic nonlinear S chr\"odinger equation with rough potential , 2024, arXiv:2403.16772

    work page arXiv 2024

  32. [32]

    Meetz , Singular potentials in nonrelativistic quantum mechanics , Il Nuovo Cimento (1955-1965), 34 (1964), pp

    K. Meetz , Singular potentials in nonrelativistic quantum mechanics , Il Nuovo Cimento (1955-1965), 34 (1964), pp. 690--708

    work page 1955

  33. [33]

    Ostermann, F

    A. Ostermann, F. Rousset, and K. Schratz , Error estimates of a F ourier integrator for the cubic S chr\" o dinger equation at low regularity , Found. Comput. Math., 21 (2021), pp. 725--765

    work page 2021

  34. [34]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari , B ose- E instein C ondensation and S uperfluidity , Oxford University Press, 2016

    work page 2016

  35. [35]

    Saqlain, T

    S. Saqlain, T. Mithun, R. Carretero-Gonz\'alez, and P. G. Kevrekidis , Dragging a defect in a droplet B ose- E instein condensate , Phys. Rev. A, 107 (2023), p. 033310

    work page 2023

  36. [36]

    Schrödinger , Quantisierung als eigenwertproblem , Annalen der Physik, 384 (1926), pp

    E. Schrödinger , Quantisierung als eigenwertproblem , Annalen der Physik, 384 (1926), pp. 361--376

    work page 1926

  37. [37]

    Schwartz, G

    T. Schwartz, G. Bartal, S. Fishman, and M. Segev , Transport and A nderson localization in disordered two-dimensional photonic lattices , Nature, 446 (2007), pp. 52--55

    work page 2007

  38. [38]

    Sulem and P.-L

    C. Sulem and P.-L. Sulem , The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse , Springer, 1999

    work page 1999

  39. [39]

    Teschl , Mathematical Methods in Quantum Mechanics , 2nd ed., Graduate Studies in Mathematics, vol

    G. Teschl , Mathematical Methods in Quantum Mechanics , 2nd ed., Graduate Studies in Mathematics, vol. 157, American Mathematical Society, Providence, RI, 2014

    work page 2014