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arxiv: 2603.02051 · v2 · submitted 2026-03-02 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.str-el· physics.comp-ph· quant-ph

Anisotropic two-dimensional magnetoexciton with exact center-of-mass separation

Dang-Khoa D. Le , Hoang-Viet Le , Dai-Nam Le , Duy-Anh P. Nguyen , Thanh-Son Nguyen , Ngoc-Tram D. Hoang , Van-Hoang Le This is my paper

Pith reviewed 2026-05-15 16:54 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-elphysics.comp-phquant-ph
keywords anisotropic 2D excitonmagnetoexcitoncenter-of-mass separationpseudomomentumblack phosphorusmagnetic fieldexciton energydiamagnetic coefficient
0
0 comments X

The pith

Exact center-of-mass separation is possible for anisotropic 2D magnetoexcitons by using conserved pseudomomentum.

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

The paper develops an exact analytical method to separate center-of-mass and relative motions for excitons in two-dimensional materials that have direction-dependent effective masses when a perpendicular magnetic field is applied. Standard approaches rely on an approximate separation that can introduce errors when electron and hole masses are similar. By starting from the complete electron-hole Hamiltonian and exploiting the conserved pseudomomentum, the authors obtain a relative-motion equation containing new anisotropy-dependent coupling terms and magnetic coefficients. These terms are absent from conventional models. The resulting equation is solved non-perturbatively for the lowest states in monolayer black phosphorus and titanium trisulfide, showing that the anisotropy couplings noticeably change the magnetic response.

Core claim

Starting from the full electron-hole Hamiltonian in a homogeneous magnetic field, the conserved pseudomomentum is used to derive a relative-motion Hamiltonian that includes previously missing anisotropy-dependent couplings and magnetic coefficients. The Schrödinger equation is then solved by the Feranchuk-Komarov operator method combined with the Levi-Civita transformation, yielding non-perturbative energies, diamagnetic coefficients, and probability densities for the ten lowest states in freestanding and encapsulated black phosphorus and titanium trisulfide across wide magnetic-field ranges.

What carries the argument

Conserved pseudomomentum, which permits exact decoupling of center-of-mass and relative coordinates without the stationary-center-of-mass approximation.

If this is right

  • Magnetoexciton energies and diamagnetic coefficients can be computed without the inaccuracies that arise from factorized wave-function approximations.
  • Anisotropy-dependent couplings alter the magnetic-field response in a manner that grows with field strength.
  • Probability densities for the lowest ten states are now available for black phosphorus and titanium trisulfide under both freestanding and hexagonal-boron-nitride-encapsulated conditions.
  • The same separation procedure applies directly to any other anisotropic two-dimensional semiconductor.

Where Pith is reading between the lines

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

  • Experimental magneto-optical spectra in anisotropic materials may require re-analysis once these extra coupling terms are included.
  • The formalism could be adapted to study how time-dependent magnetic fields affect exciton transport in the same class of materials.
  • Direct comparison of the separated energies against full-configuration-interaction calculations would quantify the size of the correction for mass ratios near unity.

Load-bearing premise

The pseudomomentum stays a good quantum number that allows exact separation even when electron and hole masses are comparable and the anisotropy is strong.

What would settle it

A direct numerical solution of the unsimplified two-particle Hamiltonian for a strongly anisotropic mass ratio that produces magnetoexciton energies differing from those obtained with the separated Hamiltonian would falsify the exact-separation result.

Figures

Figures reproduced from arXiv: 2603.02051 by Dai-Nam Le, Dang-Khoa D. Le, Duy-Anh P. Nguyen, Hoang-Viet Le, Ngoc-Tram D. Hoang, Thanh-Son Nguyen, Van-Hoang Le.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Magnetic contribution to the exciton binding energy as a function of the magnetic field [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) The probability density for a magnetoexciton in TiS [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) The probability density for a magnetoexciton in BP. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Excitons in anisotropic two-dimensional (2D) materials, defined by direction-dependent effective masses, are of pronounced interest for their roles in excitonic and magneto-optical phenomena. A perpendicular magnetic field complicates the separation of center-of-mass (c.m.) and relative motions, especially when electron and hole masses are comparable. Conventional theories often employ an approximate c.m. separation using factorized wave functions, modifying magnetic Hamiltonian terms and possibly introducing inaccuracies in magnetoexciton energy predictions. This work develops an exact analytical framework for c.m. and relative motion separation in anisotropic 2D magnetoexcitons, without resorting to the stationary-c.m. approximation. Starting from the full electron-hole Hamiltonian in a homogeneous magnetic field, the formalism uses the conserved pseudomomentum to derive a relative-motion Hamiltonian, revealing new anisotropy-dependent couplings and magnetic coefficients absent in approximate models. The resulting Schr\"odinger equation is treated via the Feranchuk-Komarov operator method and Levi-Civita transformation, allowing non-perturbative, systematically convergent solutions. Application to monolayer black phosphorus and titanium trisulfide, both freestanding and encapsulated in hexagonal boron nitride, yields magnetoexciton energies, diamagnetic coefficients, and probability densities for the ten lowest states across considerable magnetic-field ranges. The results demonstrate the significant influence of anisotropy-dependent coupling on magnetic response in systems with strong mass anisotropy. This formalism is generalizable to other anisotropic 2D semiconductors, establishing a foundation for advanced magneto-optical studies.

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 claims to derive an exact center-of-mass/relative-motion separation for anisotropic 2D magnetoexcitons starting from the full two-particle Hamiltonian in a uniform perpendicular magnetic field. Conserved pseudomomentum is used to obtain a relative-motion Schrödinger equation containing new anisotropy-dependent kinetic and magnetic coupling terms; this equation is solved non-perturbatively via the Feranchuk-Komarov operator method combined with the Levi-Civita transformation. Numerical results for the ten lowest states, diamagnetic coefficients, and probability densities are presented for monolayer black phosphorus and TiS3 (freestanding and hBN-encapsulated) over a range of magnetic fields.

Significance. If the separation is exact, the work supplies a parameter-free framework that captures anisotropy-induced couplings absent from conventional approximate treatments, thereby improving quantitative predictions of magnetoexciton spectra in strongly anisotropic 2D materials. The systematic convergence of the numerical method and the explicit demonstration of anisotropy effects on magnetic response constitute a useful advance for magneto-optical modeling.

minor comments (3)
  1. [§2.2] §2.2, after Eq. (8): the transformation to the relative coordinate under unequal masses m_e ≠ m_h and tensor effective-mass anisotropy should be written explicitly; the current notation leaves the precise definition of the reduced-mass tensor ambiguous.
  2. [Fig. 3] Fig. 3 caption: the plotted probability densities are stated to be for the ground state at B = 10 T, but the color scale and normalization convention are not specified, making quantitative comparison with other works difficult.
  3. [Table I] Table I: the diamagnetic coefficient β for the 1s state in encapsulated BP is given to three digits; adding the corresponding value obtained from the approximate c.m. method (for direct comparison) would strengthen the claim that the new terms matter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The provided summary accurately reflects the central result—an exact analytical separation of center-of-mass and relative motions for anisotropic 2D magnetoexcitons via conserved pseudomomentum—and correctly identifies the new anisotropy-dependent couplings that appear in the relative-motion Hamiltonian. We also appreciate the recognition that the Feranchuk-Komarov plus Levi-Civita approach yields systematically convergent results for the materials considered. Because the referee report lists no specific major comments, we have no point-by-point revisions to address at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the standard two-particle electron-hole Hamiltonian in a uniform perpendicular magnetic field. Conservation of pseudomomentum follows directly from the translational invariance of the uniform B field and the relative-coordinate-only Coulomb interaction; this symmetry holds independently of mass anisotropy or the m_e/m_h ratio. The relative-motion Hamiltonian is obtained by a standard change of variables using the conserved pseudomomentum, without any fitted parameters, self-referential normalizations, or ansatzes. Subsequent solution via the Feranchuk-Komarov operator method and Levi-Civita transformation are standard numerical techniques applied to the derived equation. No load-bearing steps reduce to the input by construction, and no self-citations are invoked to justify uniqueness or forbid alternatives. The central result (exact c.m. separation with anisotropy-dependent couplings) is therefore an independent consequence of the starting Hamiltonian.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the effective-mass approximation for electrons and holes in 2D crystals and the assumption of a uniform perpendicular magnetic field; no new particles or forces are introduced.

axioms (2)
  • domain assumption Effective mass approximation for electrons and holes with direction-dependent masses
    Standard modeling choice for 2D semiconductors invoked to write the initial Hamiltonian.
  • standard math Homogeneous perpendicular magnetic field
    Assumed when constructing the vector-potential terms in the Hamiltonian.

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Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    A. K. Geim and I. V. Grigorieva, Van der Waals het- erostructures, Nature499, 419 (2013)

    work page 2013

  2. [2]

    Chernikov, T

    A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2, Phys. Rev. Lett.113, 076802 (2014)

    work page 2014

  3. [3]

    G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons in atomically thin transition metal dichalco- genides, Rev. Mod. Phys.90, 021001 (2018)

    work page 2018

  4. [4]

    Laturia, M

    A. Laturia, M. L. Van de Put, and W. G. Vandenberghe, Dielectric properties of hexagonal boron nitride and tran- sition metal dichalcogenides: from monolayer to bulk, npj 2D Mater. Appl.2, 6 (2018)

    work page 2018

  5. [5]

    Z. Zhou, Y. Zhang, and Z. Zhang, Optical and electri- cal properties of two-dimensional anisotropic crystals, J. Semicond.40, 061001 (2019)

    work page 2019

  6. [6]

    Arora, Magneto-optics of layered two-dimensional semiconductors and heterostructures: Progress and prospects, J

    A. Arora, Magneto-optics of layered two-dimensional semiconductors and heterostructures: Progress and prospects, J. Appl. Phys.129, 120902 (2021)

    work page 2021

  7. [7]

    Meineke, J

    C. Meineke, J. Schlosser, M. Zizlsperger, M. Liebich, N. Nilforoushan, K. Mosina, S. Terres, A. Chernikov, Z. Sofer, M. A. Huber, M. Florian, M. Kira, F. Dirn- berger, and R. Huber, Ultrafast exciton dynamics in the atomically thin van der waals magnet crsbr, Nano Lett. 24, 4101 (2024)

    work page 2024

  8. [8]

    Lian, Y.-M

    Z. Lian, Y.-M. Li, L. Yan, L. Ma, D. Chen, T. Taniguchi, K. Watanabe, C. Zhang, and S.-F. Shi, Stark effects of Rydberg excitons in a monolayer WSe 2 P–N junction, Nano Lett.24, 4843 (2024)

    work page 2024

  9. [9]

    Okamura, S

    H. Okamura, S. Iguchi, T. Sasaki, Y. Ikemoto, T. Mori- waki, and Y. Akahama, Interband spectroscopy of lan- dau levels and magnetoexcitons in bulk black phospho- rus, Phys. Rev. B111, 125202 (2025)

    work page 2025

  10. [10]

    A. S. Rodin, A. Carvalho, and A. H. Castro Neto, Exci- tons in anisotropic two-dimensional systems, Phys. Rev. B90, 075429 (2014)

    work page 2014

  11. [11]

    J. H. Choi, P. Cui, H. Lan, and Z. Zhang, Linear scal- ing of the exciton binding energy versus the band gap of two-dimensional materials, Phys. Rev. Lett.115, 066403 (2015)

    work page 2015

  12. [12]

    H. M. Hill, A. F. Rigosi, C. Roquelet, A. Chernikov, T. C. Berkelbach, D. R. Reichman, M. S. Hybertsen, L. E. Brus, and T. F. Heinz, Observation of excitonic Rydberg states in monolayer MoS2 and WS2 by photolu- minescence excitation spectroscopy, Nano Lett.15, 2992 (2015)

    work page 2015

  13. [13]

    A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, X. Xu, and S. A. Crooker, Magnetooptics of exciton Ry- dberg states in a monolayer semiconductor, Phys. Rev. Lett.120, 057405 (2018)

    work page 2018

  14. [14]

    P. D.-A. Nguyen, D.-N. Ly, D.-N. Le, D. N.-T. Hoang, and V.-H. Le, High-accuracy energy spectra of a two- dimensional exciton screened by reduced dimensionality with the presence of a constant magnetic field, Phys. E: Low-Dimens. Syst. Nanostructures113, 152 (2019)

    work page 2019

  15. [15]

    Goryca, J

    M. Goryca, J. Li, A. V. Stier, T. Taniguchi, K. Watanabe, E. Courtade, S. Shree, C. Robert, B. Urbaszek, X. Marie, and S. A. Crooker, Revealing exciton masses and dielec- tric properties of monolayer semiconductors with high magnetic fields, Nat. Commun.10, 4172 (2019)

    work page 2019

  16. [16]

    W.-T. Hsu, J. Quan, C.-Y. Wang, L.-S. Lu, M. Camp- bell, W.-H. Chang, L.-J. Li, X. Li, and C.-K. Shih, Di- electric impact on exciton binding energy and quasipar- ticle bandgap in monolayer WS 2 and WSe 2, 2D Mater. 6, 025028 (2019)

    work page 2019

  17. [17]

    S.-Y. Chen, Z. Lu, T. Goldstein, J. Tong, A. Chaves, J. Kunstmann, L. S. R. Cavalcante, T. Wo´ zniak, G. Seifert, D. R. Reichman, T. Taniguchi, K. Watanabe, D. Smirnov, and J. Yan, Luminescent emission of excited rydberg excitons from monolayer WSe 2, Nano Lett.19, 2464 (2019)

    work page 2019

  18. [18]

    E. Liu, J. van Baren, T. Taniguchi, K. Watanabe, Y.- C. Chang, and C. H. Lui, Magnetophotoluminescence of exciton Rydberg states in monolayer WSe 2, Phys. Rev. B99, 205420 (2019). 11

    work page 2019

  19. [19]

    Ly, D.-N

    D.-N. Ly, D.-N. Le, D.-A. P. Nguyen, N.-T. D. Hoang, N.-H. Phan, H.-M. L. Nguyen, and V.-H. Le, Retrieval of material properties of monolayer transition metal dichalcogenides from magnetoexciton energy spectra, Phys. Rev. B107, 205304 (2023)

    work page 2023

  20. [20]

    Takahashi, S

    S. Takahashi, S. Kusaba, K. Watanabe, T. Taniguchi, K. Yanagi, and K. Tanaka, 3D hydrogen-like screening effect on excitons in hBN-encapsulated monolayer tran- sition metal dichalcogenides, Sci. Rep.14, 27286 (2024)

    work page 2024

  21. [21]

    Ly, D.-N

    D.-N. Ly, D.-N. Le, D.-K. D. Le, and V.-H. Le, Retrieval of fundamental material parameters of monolayer transi- tion metal dichalcogenides from experimental exciton en- ergies: An analytical approach, Phys. Rev. B112, 165417 (2025)

    work page 2025

  22. [22]

    F. Lu, Q. Hu, Y. Xu, and H. Yu, Coupled exciton internal and center-of-mass motions in two-dimensional semicon- ductors by a periodic electrostatic potential, Phys. Rev. B109, 165422 (2024)

    work page 2024

  23. [23]

    Ferreira and R

    F. Ferreira and R. M. Ribeiro, Improvements in the GW and Bethe-Salpeter-equation calculations on phos- phorene, Phys. Rev. B96, 115431 (2017)

    work page 2017

  24. [24]

    P. E. Faria Junior, M. Kurpas, M. Gmitra, and J. Fabian, k·p theory for phosphorene: Effective g-factors, Landau levels, and excitons, Phys. Rev. B100, 115203 (2019)

    work page 2019

  25. [25]

    J. Li, J. Ma, X. Cheng, Z. Liu, Y. Chen, and D. Li, Anisotropy of excitons in two-dimensional perovskite crystals, ACS Nano14, 2156 (2020)

    work page 2020

  26. [26]

    S. H. Suk, S. B. Seo, Y. S. Cho, J. Wang, and S. Sim, Ul- trafast optical properties and applications of anisotropic 2d materials, Nanophotonics13, 107 (2024)

    work page 2024

  27. [27]

    D. T.-X. Dang, D.-N. Le, and L. M. Woods, Twisting in h-bn bilayers and their angle-dependent properties, Phys. Rev. B112, 085102 (2025)

    work page 2025

  28. [28]

    Vannucci, U

    L. Vannucci, U. Petralanda, A. Rasmussen, T. Olsen, and K. S. Thygesen, Anisotropic properties of monolayer 2D materials: An overview from the C2DB database, J. Appl. Phys.128, 105101 (2020)

    work page 2020

  29. [29]

    M. N. Gjerding, A. Taghizadeh, A. Rasmussen, S. Ali, F. Bertoldo, T. Deilmann, N. R. Knøsgaard, M. Kruse, A. H. Larsen, S. Manti, T. G. Pedersen, U. Petralanda, T. Skovhus, M. K. Svendsen, J. J. Mortensen, T. Olsen, and K. S. Thygesen, Recent progress of the computa- tional 2d materials database (c2db), 2D Mater.8, 044002 (2021)

    work page 2021

  30. [30]

    V. Tran, R. Soklaski, Y. Liang, and L. Yang, Layer- controlled band gap and anisotropic excitons in few-layer black phosphorus, Phys. Rev. B89, 235319 (2014)

    work page 2014

  31. [31]

    Chaves, T

    A. Chaves, T. Low, P. Avouris, D. C ¸ akır, and F. M. Peeters, Anisotropic exciton Stark shift in black phos- phorus, Phys. Rev. B91, 155311 (2015)

    work page 2015

  32. [32]

    A. M. Wang, K. L. Seyler, V. Tran, Y. Jia, H. Zhao, H. Wang, L. Yang, X. Xu, and F. Xia, Highly anisotropic and robust excitons in monolayer black phosphorus, Nat. Nanotechnol.10, 517 (2015)

    work page 2015

  33. [33]

    J. O. Island, G. A. Steele, H. S. J. van der Zant, and A. Castellanos-Gomez, Titanium trisulfide (TiS 3): a 2D semiconductor with quasi-one-dimensional proper- ties, Sci. Rep.6, 22214 (2016)

    work page 2016

  34. [34]

    C. E. Villegas, A. S. Rodin, A. Carvalho, and A. R. Rocha, Two-dimensional exciton properties in monolayer semiconducting phosphorus allotropes, Phys. Chem. Chem. Phys.18, 27829 (2016)

    work page 2016

  35. [35]

    C. J. Paez, K. DeLello, D. Le, A. L. C. Pereira, and E. R. Mucciolo, Disorder effect on the anisotropic resistivity of phosphorene determined by a tight-binding model, Phys. Rev. B94, 165419 (2016)

    work page 2016

  36. [36]

    Torun, H

    E. Torun, H. Sahin, A. Chaves, L. Wirtz, and F. M. Peeters, Ab initio and semiempirical modeling of excitons and trions in monolayer tis 3, Phys. Rev. B98, 075419 (2018)

    work page 2018

  37. [37]

    Van der Donck and F

    M. Van der Donck and F. M. Peeters, Excitonic com- plexes in anisotropic atomically thin two-dimensional materials: Black phosphorus and TiS 3, Phys. Rev. B98, 235401 (2018)

    work page 2018

  38. [38]

    Zhang, A

    G. Zhang, A. Chaves, S. Huang, F. Wang, Q. Xing, T. Low, and H. Yan, Determination of layer-dependent exciton binding energies in few-layer black phosphorus, Sci. Adv.4, eaap9977 (2018)

    work page 2018

  39. [39]

    J. C. G. Henriques and N. M. R. Peres, Excitons in phos- phorene: A semi-analytical perturbative approach, Phys. Rev. B101, 035406 (2020)

    work page 2020

  40. [40]

    F. Wang, C. Wang, A. Chaves, C. Song, G. Zhang, S. Huang, Y. Lei, Q. Xing, L. Mu, Y. Xie, and H. Yan, Prediction of hyperbolic exciton-polaritons in monolayer black phosphorus, Nat. Comm.12, 5628 (2021)

    work page 2021

  41. [41]

    Wu, Anisotropic exciton states and excitonic absorp- tion spectra in a freestanding monolayer black phospho- rus, Phys

    S. Wu, Anisotropic exciton states and excitonic absorp- tion spectra in a freestanding monolayer black phospho- rus, Phys. E: Low-Dimens. Syst. Nanostructures.141, 115238 (2022)

    work page 2022

  42. [42]

    R. Ya. Kezerashvili, A. Spiridonova, and A. Dublin, Mag- netoexcitons in phosphorene monolayers, bilayers, and heterostructures, Phys. Rev. Research4, 013154 (2022)

    work page 2022

  43. [43]

    R. Ya. Kezerashvili and A. Spiridonova, Anisotropic magnetoexcitons in two-dimensional transition metal trichalcogenide semiconductors, Phys. Rev. Res.4, 033016 (2022)

    work page 2022

  44. [44]

    L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Black phosphorus field-effect transistors, Nat. Nanotechnol.9, 372 (2014)

    work page 2014

  45. [45]

    Avsar, I

    A. Avsar, I. J. Vera-Marun, J. Y. Tan, K. Watanabe, T. Taniguchi, A. H. Castro Neto, and B. Ozyilmaz, Air- stable transport in graphene-contacted, fully encapsu- lated ultrathin black phosphorus-based field-effect tran- sistors, ACS Nano9, 4138 (2015)

    work page 2015

  46. [46]

    S. Das, A. Sebastian, E. Pop, C. J. McClellan, A. D. Franklin, T. Grasser, T. Knobloch, Y. Illarionov, A. V. Penumatcha, J. Appenzeller, Z. Chen, W. Zhu, I. Assel- berghs, L.-J. Li, U. E. Avci, N. Bhat, T. D. Anthopou- los, and R. Singh, Transistors based on two-dimensional materials for future integrated circuits, Nat. Electron.4, 786–799 (2021)

    work page 2021

  47. [47]

    Y. Liu, X. Duan, H.-J. Shin, S. Park, Y. Huang, and X. Duan, Promises and prospects of two-dimensional transistors, Nature591, 43–53 (2021)

    work page 2021

  48. [48]

    Huang, C

    X. Huang, C. Liu, and P. Zhou, 2D semiconductors for specific electronic applications: from device to system, npj 2D Mater. Appl.6, 51 (2022)

    work page 2022

  49. [49]

    N. S. Rytova, Screened potential of a point charge in a thin film, Moscow University Physics Bulletin22, 30 (1967)

    work page 1967

  50. [50]

    L. V. Keldysh, Coulomb interaction in thin semiconduc- tor and semimetal films, JETP Lett.29, 658 (1979)

    work page 1979

  51. [51]

    J. E. Avron, I. W. Herbst, and B. Simon, Separation of center of mass in homogeneous magnetic fields, Ann. Phys.114, 431 (1978)

    work page 1978

  52. [52]

    B. R. Johnson, J. O. Hirschfelder, and K.-H. Yang, In- teraction of atoms, molecules, and ions with constant electric and magnetic fields, Rev. Mod. Phys.55, 109 12 (1983)

    work page 1983

  53. [53]

    Ruder, G

    H. Ruder, G. Wunner, H. Herold, and F. Geyer,Atoms in Strong Magnetic Fields(Springer - Verlag, Berlin, 1994)

    work page 1994

  54. [54]

    L. V. Butov, C. W. Lai, D. S. Chemla, Y. E. Lozovik, K. L. Campman, and A. C. Gossard, Observation of mag- netically induced effective-mass enhancement of quasi-2d excitons, Phys. Rev. Lett.87, 216804 (2001)

    work page 2001

  55. [55]

    Ly, D.-N

    D.-N. Ly, D.-N. Le, N.-H. Phan, and V.-H. Le, Ther- mal effect on magnetoexciton energy spectra in mono- layer transition metal dichalcogenides, Phys. Rev. B107, 155410 (2023)

    work page 2023

  56. [56]

    L. D. Landau and E. M. Lifshitz,Quantum Mechanics: Non-relativistic Theory(Pergamon, Oxford, 1977)

    work page 1977

  57. [57]

    Galiautdinov, Anisotropic keldysh interaction, Phys

    A. Galiautdinov, Anisotropic keldysh interaction, Phys. Lett. A383, 3167 (2019)

    work page 2019

  58. [58]

    I. D. Feranchuk and L. I. Komarov, The operator method of the approximate solution of the Schr¨ odinger equation, Phys. Lett. A88, 211 (1982)

    work page 1982

  59. [59]

    Feranchuk, A

    I. Feranchuk, A. Ivanov, V. H. Le, and A. Ulya- nenkov,Non-perturbative Description of Quantum Sys- tems, Springer Tracts in Modern Physics, Vol. 894 (Springer International Publishing, Cham, 2015)

    work page 2015

  60. [60]

    L. V. Hoang and N. T. Giang, The algebraic method for two-dimensional quantum atomic systems, J. Phys. A: Math. Gen.26, 1409 (1993)

    work page 1993

  61. [61]

    LAPACK: Linear Algebra PACKage, Subrou- tine dsygvx.f

    Netlib.org. LAPACK: Linear Algebra PACKage, Subrou- tine dsygvx.f

    work page