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REVIEW 3 major objections 5 minor 46 references

Spectral Analysis of the Schr\"odinger Operator for the Incommensurate System

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A higher-dimensional embedding plus a tiny smoothing term lets the spectrum of a Schrödinger operator with two incommensurate periodic potentials be approximated by the spectra of ordinary periodic operators.

desk verdict A plausible spectral-equivalence and regularization framework for incommensurate Schrödinger operators, but the central approximation theorem only proves one inclusion and rests on an unproved smoothing claim. read the letter →

arxiv 2602.08308 v2 pith:4XYRK6JU submitted 2026-02-09 math-ph math.MP

Yan Li , Yujian Song , Aihui Zhou This is my paper · ORCID
classification math-phmath.MP MSC 35J1035J7047A1081Q10
keywords incommensuratesystemquasi-periodicSchrödingeroperatorspectralapproximationregularizationBloch-typesolutiondegenerateellipticBloch-Floquettransformself-adjointextension
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

This paper proves that the spectrum of a Schrödinger operator with two incommensurate periodic potentials—the standard model of twisted or moiré layered materials—can be approximated by the spectra of a family of regularized operators that are uniformly elliptic and periodic. The route is to lift the problem to a doubled space, where the extended operator is periodic but degenerate; the paper constructs its unique self-adjoint extension, shows that its spectrum coincides with the original one when the potentials are $C^1$, and then proves that adding a small correction term proportional to the squared difference of the gradients makes strong resolvent convergence hold as the regularization parameter goes to zero. As a corollary, every spectral point of the incommensurate operator is the limit of spectral points of the regularized periodic model, and generalized eigenfunctions are limits of Bloch waves. A sympathetic reader would care because it supplies a rigorous justification for using standard periodic numerical algorithms on incommensurate systems, with the regularization parameter as a controlled approximation parameter.

What carries the argument

The central object is the regularized extended operator $\tilde{H}^\delta := \tilde{H} - (\delta/2) \sum_i (\partial_{r_i} - \partial_{r'_i})^2$ on $L^2(R^{2d})$, where $\tilde{H} = -1/2 \sum_i (\partial_{r_i} + \partial_{r'_i})^2 + V_1(r) + V_2(r')$ is the extended operator obtained by embedding the two incommensurate lattices into separate copies of $R^d$. Without the $\delta$ term, $\tilde{H}$ is degenerate elliptic and its naive domain $H^2(R^{2d})$ does not make it self-adjoint; the paper takes the unique self-adjoint closure (domain $G$). The $\delta$ term changes the principal symbol from a rank-deficient quadratic form to a uniformly elliptic one, so the regularized operator is self-adjoint on $H^2(R^{2d})$, admits a Bloch-Floquet direct integral decomposition, and each fiber ha

What would settle it

Solve the resolvent equation $(iI - \tilde{H}(k))u = f$ for a smooth $f$ on the torus: if for some smooth $f$ the solution $u$ is not in $H^2$, the smoothing assertion fails and the spectral approximation theorem loses its foundation. Alternatively, for a concrete incommensurate pair (e.g., cosine potentials with golden-ratio frequencies), numerically check whether the Hausdorff distance between $\sigma(\tilde{H}^\delta)$ and $\sigma(H)$ tends to zero as $\delta\to 0^+$.

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Extended reading notes

Core claim

The central discovery is a spectral approximation theorem: for every $\lambda$ in $\sigma(H)$ there is a sequence $\lambda^\delta \in \sigma(\tilde{H}^\delta)$ with $\lambda^\delta \to \lambda$ as $\delta\to 0^+$, where $\tilde{H}^\delta$ are elliptic, periodic operators in doubled dimension. The proof shows that the spectrum of the original incommensurate operator equals the spectrum of the self-adjoint closure of the degenerate extended operator, and that the regularized operators converge to that closure in strong resolvent sense. This makes the quasi-periodic spectrum, which lacks any Bloch reduction, a limit of ordinary periodic (Bloch) spectra. The paper further shows the solution set of the incommensurate Schrödinger equation is non-empty at every spectral point a

Load-bearing premise

The load-bearing premise is that the resolvent of the degenerate extended operator is smoothing on the torus (smooth inputs give smooth outputs) even though the operator is not elliptic; without that, the $H^2$ estimate that drives the approximation is unsupported.

Editorial extensions

If this is right

  • The spectrum of a twisted-bilayer-type Schrödinger operator can be computed to arbitrary accuracy by diagonalizing a family of standard periodic (elliptic) operators on a torus.
  • Bloch's theorem and numerical algorithms for periodic systems become applicable, with δ as a controllable regularization parameter.
  • The spectral equivalence σ(H)=σ(\tilde H) identifies the regime in which the higher-dimensional embedding is exact: potentials of class C^1; with merely continuous potentials only one inclusion is claimed.
  • Generalized eigenfunctions of the incommensurate operator exist at every spectral point and are limits of Bloch waves of the regularized model on compact sets.
  • The argument is stated for d=1,2 and two layers, but the construction extends in principle to higher dimensions and more stacked layers.

Reading between the lines

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

  • If the approximation is quantitative, the convergence rate as δ→0+ should depend on the Diophantine properties of the incommensurate ratio; deriving such a rate is a natural next step not addressed in the paper.
  • The same two-step recipe—self-adjoint closure then elliptic regularization—should transfer to other non-elliptic periodic operators, for example models with constraints other than r=r'.
  • A testable prediction is that small-δ eigenvalues of \tilde H^δ reproduce flat bands and van Hove singularities of twisted bilayer models, providing a supercell-free computational route to moiré physics.
  • The existence of approximating Bloch-type eigenfunctions suggests that time-dependent Schrödinger problems for incommensurate systems can be approached by evolving the regularized periodic model and letting δ→0.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the Schrödinger operator H = -1/2 Δ + V_1(r) + V_2(r) on L^2(R^d) for a bilayer incommensurate system, where V_1, V_2 are periodic with incommensurate lattices. The authors embed the system into R^{2d} via coordinates (r,r') and introduce the extended operator \tilde H = -1/2 ∑(∂_{r_i}+∂_{r'_i})^2 + V_1(r)+V_2(r'), then construct a self-adjoint extension. To overcome the degeneracy of \tilde H, they introduce a family of elliptic regularized operators \tilde H^δ = \tilde H - (δ/2)∑(∂_{r_i}-∂_{r'_i})^2. The main claims are: (i) σ_G(\tilde H) = σ(H) under V_j ∈ C^1; (ii) for every λ ∈ σ(H) there is λ^δ ∈ σ(\tilde H^δ) with λ^δ → λ; (iii) every λ ∈ σ(H) admits a nontrivial Bloch-type solution that is approximated by Bloch solutions of the regularized models. The argument combines Bloch-Floquet theory, direct integral decompositions, Fredholm theory, and a diagonal ergodic averaging argument.

Significance. If the main theorems were fully established, the paper would provide a useful rigorous framework for approximating spectra and generalized eigenfunctions of incommensurate Schrödinger operators by spectra of standard periodic elliptic operators, thereby legitimizing a class of numerical methods for twisted bilayer systems. The paper is clearly structured and contains several correct and useful ingredients: the self-adjoint extension of the degenerate extended operator, the direct integral decomposition of the regularized operators, and the identification of Fredholm/discrete spectra for each δ > 0. However, as discussed below, the proof of the central approximation theorem contains a gap, and the Bloch-type solution theorem has an additional nontrivial issue. These problems are load-bearing and must be repaired before the paper can be accepted.

major comments (3)
  1. [Theorem 4.7, §4.2, Eq. (4.7)] The proof of strong resolvent convergence relies on the assertion 'R(i;\tilde H(\tilde k))f ∈ C^∞(T^{2d})' for f ∈ C^∞. This is neither proved nor true under the standing assumption V_j ∈ C^0. In coordinates p=(r+r')/2, q=(r-r')/2, the operator \tilde H(\tilde k) contains no q-derivatives; it is of the form -1/2 Δ_p + V_1(p+q)+V_2(p-q) plus first-order terms. The resolvent therefore does not smooth in q: any q-regularity of R(i;\tilde H(\tilde k))f must come from the potential, and for merely continuous V the q-derivatives of the resolvent need not exist. Consequently the bound \|R(i;\tilde H(\tilde k))f\|_{H^2(T^{2d})} used in (4.7) is not available. Since (4.7) is the only step in the proof that yields strong resolvent convergence, and since Corollary 4.8 and Theorem 5.3 both depend on Theorem 4.7, the main spectral approximation theorem is not established as written. The proof could b
  2. [Theorem 5.2(b), §5.1; Theorem 5.5, §5.2] In the compactness argument, the authors normalize the regularized eigenfunctions \tilde v^δ in H^3 (or H^2 in Theorem 5.5), pass to a limit \tilde v_* ∈ H^s(T^{2d}), and define u_*(r) = \tilde v_*(r,r) by restriction to the diagonal. No argument shows that this diagonal restriction is nonzero. For a nonzero Sobolev function on T^{2d}, the trace on the diagonal can vanish identically (e.g., functions with Fourier support in n-m vanish on r=r'). If u_* = 0, then the Weyl sequence ψ_R = χ_R u_* in Eq. (5.12)–(5.13) has zero denominator, so the proof that σ_G(\tilde H) ⊂ σ(H) collapses. Likewise, Theorem 5.5(a) asserts Θ_0(λ) ≠ {0}, but the constructed u_* may be zero. The authors need an additional normalization or a lower bound on the diagonal trace of the approximating eigenfunctions, or a different construction of the limiting state. This is not a cosmetic point: regularized eigenfuncti
  3. [Corollary 4.8, §4.2] The diagonal argument at the end of the proof is not rigorous as written. The proof fixes ε > 0, obtains \tilde λ_ε ∈ σ(\tilde H(\tilde k_ε)), and then invokes Theorem 4.7 to get \tilde λ^δ_ε for δ > 0. The final choice '\tilde λ^δ := \tilde λ^δ_δ' requires a simultaneous limit in δ and ε; without additional estimates, convergence of \tilde λ^δ_δ to \tilde λ is not justified. This is a standard diagonal-extraction issue and can likely be repaired, but the current text does not supply the needed argument.
minor comments (5)
  1. [Theorem 5.2(a), §5.1, Eq. (5.2)–(5.4)] The norm computation for ψ_n is incorrect: with p=(r+r')/2, q=(r-r')/2, the Jacobian is 2^d, so ∥ψ_n∥_{L^2(R^{2d})} = 2^{d/2}, not (1/2)^{2d-1}. Since only boundedness away from zero is needed, the argument can be fixed by renormalization, but the displayed formula should be corrected.
  2. [Theorem 5.2(b), §5.1] The statement 'standard Schauder estimates ... imply \tilde v^δ ∈ C^3(T^{2d})' is overstated for V_j ∈ C^1. Directly, elliptic regularity with C^1 coefficients gives at most C^{2,α} or H^3 after bootstrapping; C^3 is not automatic, especially in dimension 2d=4. Since the proof only needs H^3 bounds and a Sobolev-trace argument, this can be repaired by replacing C^3 with H^3 and making the bootstrap explicit.
  3. [Definition 4.1, §4.1] There is a typo: '\tilde H^δ : L^2(R^{2d}) → L^2(R)' should read '→ L^2(R^{2d})'.
  4. [§4.1, after Proposition 4.4] The statement that 'the spectrum of the regularized operator \tilde H^δ is absolutely continuous' is too strong. Continuity of each band function \tilde λ^δ_j(\tilde k) implies that the spectrum is a union of intervals, but it does not by itself establish absolute continuity of the spectral measure. This is not essential to the main approximation claim, but the wording should be adjusted.
  5. [General presentation] The manuscript contains numerous small typos and notational inconsistencies (e.g., 'Schr¨odinger', 'L2(R)' in Definition 4.1, and the phrase in Proposition 3.1 'as an operator L2(R^{2d}) → L2(R^{2d}) on the domain H^2(R^{2d}) is self-adjoint' where the intended statement is that multiplication by V is a bounded self-adjoint perturbation). A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral approximation theorem is proved from standard spectral theory, not from its own conclusions.

full rationale

The derivation chain is not circular. The central claim (Theorem 5.3) is obtained via independent, in-paper arguments: Theorem 5.2 proves spectral equivalence between H and the self-adjoint extension of the embedded operator via Weyl sequences and limiting regularized eigenfunctions; Section 3 gives the Bloch-Floquet direct integral decomposition; Theorem 4.7 proves strong resolvent convergence of the elliptic regularized operators to the degenerate fiber operators; and Lemma 4.6 converts strong resolvent convergence into spectral approximation. These use standard tools (Kato-Rellich, Rellich-Kondrachov, Riesz-Schauder theory, unique ergodicity), not the target theorem. The self-citations [46], [41], [40], [42], [33], and [18] are used for provenance of the embedding, numerical methodology, and Bloch theorem background, but the load-bearing spectral results are reproved in the text and do not reduce to those citations. The potential concern about Theorem 4.7, namely the assertion that R(i; ~H(k)) maps C^∞ to C^∞ for merely continuous potentials, is a regularity gap in the proof rather than a circular reduction: it does not equate an output with an input by construction, and it is not resolved by citing the authors' own prior work. No fitted parameter is renamed as a prediction, and no claimed result is definitionally identical to an assumption, so the appropriate finding is no circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No fitted parameters or physical new entities. The central claims rest on standard spectral theory, a regularity assumption on the potential, an unproved smoothness assertion for the degenerate resolvent, and unique ergodicity of linear flows.

assumptions (4)
  • domain assumption Vj ∈ C^0(Γj) (or C^1 for part b) and R1,R2 incommensurate
    The proof of σ(H)⊂σ_G(\tilde H) uses continuity of V; the reverse inclusion and Bloch-type solution regularity require C^1.
  • ad hoc to paper R(i;\tilde H(\tilde k)) maps C∞ to C∞ for degenerate elliptic \tilde H
    Used in Theorem 4.7 (equation 4.7) without proof; if false, the strong resolvent convergence argument collapses.
  • standard math Unique ergodicity of the diagonal linear flow on the torus T^{2d}
    Applied in Theorem 5.2(b) to get the Weyl sequence via ball averages; standard for incommensurate linear flows.
  • standard math Standard elliptic regularity and Rellich-Kondrachov compactness
    Used to establish discreteness of regularized spectra and compactness of eigenfunction sequences.
invented entities (1)
  • Regularized operator \tilde H^δ
    purpose: Restore ellipticity and periodicity so classical Bloch theory and numerical methods apply
    A mathematical construction; its main support is the approximation theorem proved in this paper, so no independent external evidence.

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Pith. "Pith review of Spectral Analysis of the Schr\"odinger Operator for the Incommensurate System." pith.science (2026). https://pith.science/paper/4XYRK6JU

@misc{pith2026260208308,
  author       = {Pith},
  title        = {Pith review of: Spectral Analysis of the Schr\"odinger Operator for the Incommensurate System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4XYRK6JU}},
  note         = {Machine review of arXiv:2602.08308}
}
read the original abstract

Many novel and unique physical phenomena in incommensurate systems can be illustrated and predicted using their spectral structure and electronic state distributions. However, the absence of periodicity in these systems poses significant challenges for obtaining the associated information. In this paper, by embedding the system into higher dimensions together with introducing a regularization technique, we prove that the spectrum of the Schr\"odinger operator for the incommensurate system can be approximated by the spectra of a family of regularized Schr\"odinger operators, which are elliptic, retain periodicity, and enjoy favorable analytic and spectral properties. We also show the well-posedness of the probability density describing the electronic state distribution of the incommensurate system, which can be approximated by the ones generated by the Bloch solutions to the regularized model. Our analysis provides theoretical support for understanding and computing incommensurate systems.

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Works this paper leans on

46 extracted references

  1. [1]

    R. A. Adams and J. J. Fournier , Sobolev Spaces, Elsevier, Amsterdam, 2003

    2003

  2. [2]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist , Nat. Mater., 19 (2020), pp. 1265–1275

    2020

  3. [3]

    A vron and B

    J. A vron and B. Simon, Almost periodic Schr¨ odinger operators: I. limit periodic potentials, Commun. Math. Phys., 82 (1981), pp. 101–120

    1981

  4. [4]

    A vron and B

    J. A vron and B. Simon, Almost periodic Schr¨ odinger operators: II. the integrated density of states, Duke Math. J., 50 (1983), pp. 369–391

    1983

  5. [5]

    Britnell, R

    L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y.- J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, et al. , Strong light-matter interactions in heterostructures of atomically thin films , Science, 340 (2013), pp. 1311– 1314

    2013

  6. [6]

    Canc `es, L

    ´E. Canc `es, L. Garrigue, and D. Gontier , Simple derivation of moir´ e-scale continuous models for twisted bilayer graphene , Phys. Rev. B, 107 (2023), p. 155403

    2023

  7. [7]

    Y. Cao, V. F atemi, A. Demir, S. F ang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez- Yamagishi, K. W atanabe, T. Taniguchi, E. Kaxiras, et al. , Correlated insulator be- haviour at half-filling in magic-angle graphene superlattices , Nature, 556 (2018), pp. 80–84. 22

    2018

  8. [8]

    Y. Cao, V. F atemi, S. F ang, K. W atanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo- Herrero, Unconventional superconductivity in magic-angle graphene superlattices , Na- ture, 556 (2018), pp. 43–50

    2018

Show all 46 references
  1. [9]

    S. Carr, D. Massatt, S. F ang, P. Cazeaux, M. Luskin, and E. Kaxiras , Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle , Phys. Rev. B, 95 (2017), p. 075420

    2017

  2. [10]

    H. Chen, A. Zhou, and Y. Zhou, Plane wave study on the localized-extended transition in the one-dimensional incommensurate systems , Comput. Mater. Sci., 188 (2021), p. 110242

    2021

  3. [11]

    Corwin and F

    L. Corwin and F. P. Greenleaf , Representations of Nilpotent Lie Groups and Their Appli- cations: Volume 1, Part 1, Basic Theory and Examples , vol. 18, Cambridge University Press, Cambridge, 1990

    1990

  4. [12]

    X. Dai, A. Zhou, and Y. Zhou , An extended plane wave framework for the electronic struc- ture calculations of twisted bilayer material systems , Comput. Mater. Sci., 226 (2023), p. 112204

    2023

  5. [13]

    C. R. De Oliveira , Intermediate Spectral Theory and Quantum Dynamics , vol. 54, Springer Science & Business Media, Berlin, 2008

    2008

  6. [14]

    C. R. Dean, L. W ang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, et al. , Hofstadter’s butterfly and the fractal quan- tum hall effect in moir´ e superlattices, Nature, 497 (2013), pp. 598–602

    2013

  7. [15]

    E. I. Dinaburg and Y. G. Sinai , The one-dimensional schrodinger equation with a quasiperi- odic potential, Funct. Anal. Appl., 9 (1975), pp. 8–21

    1975

  8. [16]

    S. S. Dindorkar, A. S. Kurade, and A. H. Shaikh, Magical moir´ e patterns in twisted bilayer graphene: A review on recent advances in graphene twistronics , Chem. Phys. Impact, 7 (2023), p. 100325

    2023

  9. [17]

    Ding and A

    J. Ding and A. Zhou , Statistical Properties of Deterministic Systems , Springer Science & Business Media, Berlin, 2010

    2010

  10. [18]

    Z. Ding, Y. Li, A. Zhou, and Y. Zhou , A regularization based computational method for the quantum incommensurate problems, In preparation, (2026)

    2026

  11. [19]

    Eliasson , Floquet solutions for the 1-dimensional quasi-periodic schr¨ odinger equation, Commun

    L.-H. Eliasson , Floquet solutions for the 1-dimensional quasi-periodic schr¨ odinger equation, Commun. Math. Phys., 146 (1992), pp. 447–482

    1992

  12. [20]

    Etter, D

    S. Etter, D. Massatt, M. Luskin, and C. Ortner , Modeling and computation of kubo conductivity for two-dimensional incommensurate bilayers , Multiscale Model. Simul., 18 (2020), pp. 1525–1564

    2020

  13. [21]

    G. B. Folland, A Course in Abstract Harmonic Analysis , CRC press, Boca Raton, 2016

    2016

  14. [22]

    A. K. Geim and I. V. Grigorieva , Van der waals heterostructures , Nature, 499 (2013), pp. 419–425

    2013

  15. [23]

    Gontier and S

    D. Gontier and S. Lahbabi, Convergence rates of supercell calculations in the reduced hartree- fock model, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 1403–1424

    2016

  16. [24]

    C.-Y. Hao, Z. Zhan, P. A. Pantale´on, J.-Q. He, Y.-X. Zhao, K. W atanabe, T. Taniguchi, F. Guinea, and L. He, Robust flat bands in twisted trilayer graphene moir´ e quasicrystals, Nat. Commun., 15 (2024), p. 8437

    2024

  17. [25]

    H¨ormander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Springer Science & Business Media, Berlin, 2007

    L. H¨ormander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Springer Science & Business Media, Berlin, 2007

    2007

  18. [26]

    Karpeshina and R

    Y. Karpeshina and R. Shterenberg , Extended States for the Schr¨ odinger Operator with Quasi-periodic Potential in Dimension Two , vol. 258, American Mathematical Society, Providence, 2019

    2019

  19. [27]

    Kato , Perturbation Theory for Linear Operators , vol

    T. Kato , Perturbation Theory for Linear Operators , vol. 132, Springer Science & Business Media, Berlin, 2013

    2013

  20. [28]

    D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. Basov, A. N. Pasupathy, and A. Rubio, Moir´ e heterostructures as a condensed-matter quantum simulator , Nat. Phys., 17 (2021), pp. 155–163

    2021

  21. [29]

    D. S. Koda, F. Bechstedt, M. Marques, and L. K. Teles , Coincidence lattices of 2d crystals: Heterostructure predictions and applications , J. Phys. Chem. C, 120 (2016), pp. 10895–10908

    2016

  22. [30]

    Komsa and A

    H.-P. Komsa and A. V. Krasheninnikov , Electronic structures and optical properties of realistic transition metal dichalcogenide heterostructures from first principles , Phys. Rev. B, 88 (2013), p. 085318

    2013

  23. [31]

    Kuchment , Floquet Theory for Partial Differential Equations , vol

    P. Kuchment , Floquet Theory for Partial Differential Equations , vol. 60, Birkh¨ auser, Basel, 1993

    1993

  24. [32]

    Kuchment , An overview of periodic elliptic operators , Bull

    P. Kuchment , An overview of periodic elliptic operators , Bull. Am. Math. Soc., 53 (2016), pp. 343–414

    2016

  25. [33]

    Y. Li, B. Yang, and A. Zhou, A mathematical aspect of Bloch’s theorem, Phys. Scr., 99 (2024), 23 p. 105263

    2024

  26. [34]

    Loh and R

    G. Loh and R. Pandey , A graphene–boron nitride lateral heterostructure–a first-principles study of its growth, electronic properties, and chemical topology , J. Mater. Chem. C, 3 (2015), pp. 5918–5932

    2015

  27. [35]

    Massatt, S

    D. Massatt, S. Carr, M. Luskin, and C. Ortner , Incommensurate heterostructures in momentum space, Multiscale Model. Simul., 16 (2018), pp. 429–451

    2018

  28. [36]

    Massatt, M

    D. Massatt, M. Luskin, and C. Ortner , Electronic density of states for incommensurate layers, Multiscale Model. Simul., 15 (2017), pp. 476–499

    2017

  29. [37]

    O. A. Nielsen , Direct Integral Theory, CRC Press, Boca Raton, 2020

    2020

  30. [38]

    Reed and B

    M. Reed and B. Simon , Methods of Modern Mathematical Physics: Functional Analysis , vol. 1, Gulf Professional Publishing, Amsterdam, 1980

    1980

  31. [39]

    W alters, An Introduction to Ergodic Theory , vol

    P. W alters, An Introduction to Ergodic Theory , vol. 79, Springer Science & Business Media, Berlin, 2000

    2000

  32. [40]

    W ang, H

    T. W ang, H. Chen, A. Zhou, and Y. Zhou , Layer-splitting methods for time-dependent schr¨ odinger equations of incommensurate systems, Commun. Comput. Phys., 30 (2021), pp. 1474–1498

    2021

  33. [41]

    W ang, H

    T. W ang, H. Chen, A. Zhou, Y. Zhou, and D. Massatt , Convergence of the planewave approximations for quantum incommensurate systems, Multiscale Model. Simul., 23 (2025), pp. 545–576

    2025

  34. [42]

    W ang, Y

    T. W ang, Y. Zhou, and A. Zhou, Localization in the incommensurate systems: A plane wave study via effective potentials , Commun. Comput. Phys., 35 (2024), pp. 1029–1044

    2024

  35. [43]

    A. B. W atson, T. Kong, A. H. MacDonald, and M. Luskin, Bistritzer–macdonald dynamics in twisted bilayer graphene , J. Math. Phys., 64 (2023), p. 031502

    2023

  36. [44]

    Woods, L

    C. Woods, L. Britnell, A. Eckmann, R. Ma, J. Lu, H. Guo, X. Lin, G. Yu, Y. Cao, R. V. Gorbachev, et al., Commensurate–incommensurate transition in graphene on hexagonal boron nitride, Nat. Phys., 10 (2014), pp. 451–456

    2014

  37. [45]

    M. Xu, T. Liang, M. Shi, and H. Chen , Graphene-like two-dimensional materials , Chem. Rev., 113 (2013), pp. 3766–3798

    2013

  38. [46]

    Y. Zhou, H. Chen, and A. Zhou , Plane wave methods for quantum eigenvalue problems of incommensurate systems, J. Comput. Phys., 384 (2019), pp. 99–113. 24

    2019

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