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arxiv: 2602.07281 · v2 · submitted 2026-02-07 · 🪐 quant-ph · cond-mat.quant-gas· nlin.PS· physics.optics

Recognition: 2 theorem links

· Lean Theorem

The continuous spectrum of bound states in expulsive potentials

H. Sakaguchi , B.A. Malomed , A.C. Aristotelous , E.G. Charalampidis
Authors on Pith no claims yet

Pith reviewed 2026-05-16 06:54 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gasnlin.PSphysics.optics
keywords expulsive potentialSchrodinger equationbound states in the continuumnormalizable statescontinuous spectrumvortex statesself-trapping
0
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The pith

Steep expulsive potentials support normalizable eigenstates forming continuous spectra in 1D and 2D Schrödinger equations

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

The paper demonstrates that expulsive potentials steeper than quadratic ones in the Schrödinger equation produce normalizable eigenstates, contrary to the intuition that they should delocalize states. These states form full continuous spectra in both one and two dimensions. In one dimension the eigenstates are even or odd, while in two dimensions they can have arbitrary vorticity. Analytic asymptotic expressions for the wave functions match numerical results, and exact solutions exist for certain vortex states. This finding extends the idea of bound states in the continuum to these linear settings and briefly considers nonlinear extensions.

Core claim

In one- and two-dimensional Schrödinger equations, expulsive potentials steeper than quadratic give rise to normalizable eigenstates that constitute full continuous spectra. The states are spatially even and odd in 1D and may carry any vorticity in 2D. Asymptotic wave function expressions derived analytically agree with numerical solutions, and special exact vortex solutions are obtained in 2D.

What carries the argument

The expulsive potential steeper than quadratic, which produces decaying asymptotic tails in the eigenfunctions allowing normalizability over a continuous energy range.

If this is right

  • In 1D, both even and odd parity states exist for any energy value.
  • In 2D, states with any integer vorticity are supported.
  • Exact solutions for vortex states are available in 2D.
  • The nonlinear extension via Gross-Pitaevskii shows slight deformation but maintained stability in 1D.

Where Pith is reading between the lines

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

  • If confirmed, this could allow trapping mechanisms in purely repulsive linear potentials for quantum particles.
  • Similar effects might appear in optical waveguides with engineered refractive index profiles.
  • Extensions to time-dependent or higher-dimensional cases could be tested numerically.

Load-bearing premise

The potential must be exactly of the expulsive form and strictly steeper than quadratic, without perturbations that would prevent the required asymptotic decay.

What would settle it

Solving the eigenvalue problem numerically for a potential like V(x) = -|x|^{2+epsilon} and checking if the wave function remains square-integrable at infinity for a range of energies.

Figures

Figures reproduced from arXiv: 2602.07281 by A.C. Aristotelous, B.A. Malomed, E.G. Charalampidis, H. Sakaguchi.

Figure 1
Figure 1. Figure 1: FIG. 1. The wave function [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The continuous curve: the numerically found spatial [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The continuous curve: the numerically found spatial [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. In panels (a) and (b), the continuous curves represen [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The spatially even eigenstate produced by the numeri [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The numerically found dependence of the coordinate [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) The stable evolution of the spatially even eigens [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The stable spatially odd eigenstate produced by the n [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) The absolute value [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

On the contrary to the common intuition, which suggests that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schroedinger equations, which include expulsive potentials that are steeper than the quadratic ones, give rise to normalizable eigenstates, which may be considered as a manifestation of effective self-trapping in the linear system. These states constitute full continuous spectra in both the 1D and 2D cases. In 1D, they are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic expressions for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with full numerical solutions. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are briefly considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability

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

1 major / 1 minor

Summary. The manuscript claims that 1D and 2D Schrödinger equations with expulsive potentials steeper than quadratic (V ∼ −|x|^α, α>2) admit normalizable eigenstates that form continuous spectra. These states are spatially even/odd in 1D and carry arbitrary vorticity in 2D; asymptotic analytic expressions for the wave functions are derived and stated to agree with numerical solutions, with exact vortex solutions provided in 2D. The results are presented as an extension of bound states in the continuum, with brief discussion of the nonlinear Gross-Pitaevskii extension.

Significance. If the central claim holds, the work would be significant for demonstrating effective self-trapping in linear systems with steeply expulsive potentials, challenging standard intuition about delocalization, and extending the bound-states-in-the-continuum concept to continuous spectra in both quantum mechanics and paraxial optics. The availability of analytic asymptotics, numerical agreement, and exact 2D vortex solutions would strengthen the contribution.

major comments (1)
  1. [Abstract and asymptotic analysis sections] The purported eigenfunctions satisfy the Schrödinger ODE pointwise and are normalizable, but for V ∼ −|x|^α (α>2) the asymptotic |ψ| ∼ |x|^{-α/4} implies |Vψ| ∼ |x|^{3α/4} so that ∫|Vψ|^2 dx diverges at infinity. Consequently ψ lies outside the domain of the self-adjoint operator H = −∇² + V (where Vψ must belong to L²), and cannot be an eigenstate in the L² Hilbert-space sense. This directly undermines the central claim that these states “constitute full continuous spectra” of eigenstates; the issue must be resolved by showing membership in dom(H), by adopting a quadratic-form definition, or by clarifying the precise sense in which “eigenstate” is used.
minor comments (1)
  1. [Abstract] The abstract states that the cubic nonlinearity “slightly deforms the eigenstates, maintaining their stability” in 1D; a brief quantitative illustration (e.g., overlap or norm deviation) would strengthen this statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the thorough review and for highlighting the important issue concerning the domain of the Hamiltonian operator. This comment has prompted us to revise the manuscript to provide a clearer explanation of the mathematical framework in which our results are presented.

read point-by-point responses

  1. Referee: [Abstract and asymptotic analysis sections] The purported eigenfunctions satisfy the Schrödinger ODE pointwise and are normalizable, but for V ∼ −|x|^α (α>2) the asymptotic |ψ| ∼ |x|^{-α/4} implies |Vψ| ∼ |x|^{3α/4} so that ∫|Vψ|^2 dx diverges at infinity. Consequently ψ lies outside the domain of the self-adjoint operator H = −∇² + V (where Vψ must belong to L²), and cannot be an eigenstate in the L² Hilbert-space sense. This directly undermines the central claim that these states “constitute full continuous spectra” of eigenstates; the issue must be resolved by showing membership in dom(H), by adopting a quadratic-form definition, or by clarifying the precise sense in which “eigenstate” is used.

    Authors: We agree with the referee that the wave functions, while satisfying the Schrödinger equation pointwise and being square-integrable, do not lie in the domain of the standard self-adjoint Hamiltonian because Vψ ∉ L². In the revised version of the manuscript, we have clarified this point by specifying that the states are understood as normalizable solutions to the differential equation, rather than as eigenfunctions in the strict sense of the unbounded operator on L². We have added a paragraph in the introduction and in the discussion section explaining that in the context of this work, 'eigenstates' refers to solutions of the ODE that are normalizable, and we discuss the implications for the spectrum in both quantum mechanics and optics. We have also noted that for applications in paraxial optics, the equation is solved directly as a PDE without requiring the operator domain. This revision addresses the concern without altering the core findings, which remain valid in the classical sense. We believe this resolves the issue by adopting the clarification approach suggested by the referee. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via direct asymptotic and numerical solution of the Schrödinger equation

full rationale

The paper obtains its central results by solving the time-independent Schrödinger equation for the specified expulsive potentials, deriving asymptotic wave-function forms far from the origin through standard WKB-style or direct substitution analysis, and confirming agreement with numerical integration. No parameters are fitted to a data subset and then re-presented as predictions, no self-citations supply the load-bearing uniqueness or ansatz, and the normalizability claim follows directly from the exhibited decay rates without redefinition. The derivation therefore stands on independent first-principles content rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard time-independent Schrödinger equation and the assumption that the potential is purely expulsive and steeper than quadratic. No free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math The time-independent Schrödinger equation governs the eigenstates
    Standard framework for linear quantum mechanics
  • domain assumption The potential is expulsive and steeper than quadratic
    Explicitly required for the normalizable continuous spectrum to appear

pith-pipeline@v0.9.0 · 5544 in / 1269 out tokens · 38749 ms · 2026-05-16T06:54:19.541700+00:00 · methodology

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Zettl A., Sturm–Liouville Theory (AMS, Providence, 2005)

    work page 2005

  2. [2]

    Lin J. S. and Zhang H., Mathematical theory for topologic al photonic materials in one dimension, J. Phys. A: Math. Gen . 55, 495203 (2022); DOI 10.1088/1751-8121/aca9a5

    work page doi:10.1088/1751-8121/aca9a5 2022

  3. [3]

    F., Meng G., and Zhang Z

    Chu J. F., Meng G., and Zhang Z.. Minimizations of positiv e periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems, Advances in Mathematics 432, 109272 (2023), https://doi.org/10.1016/j.aim.2023.10 9272

    work page doi:10.1016/j.aim.2023.10 2023

  4. [4]

    V., Spectrum completion and inverse Sturm -Liouville problems, Mathematical Methods in the Applied Sciences 46, 5821-5835 (2023), https://doi.org/10.1002/mma.8869

    Kravchenko V. V., Spectrum completion and inverse Sturm -Liouville problems, Mathematical Methods in the Applied Sciences 46, 5821-5835 (2023), https://doi.org/10.1002/mma.8869

    work page doi:10.1002/mma.8869 2023

  5. [5]

    Landau L. D. and Lifshitz E. M., Quantum Mechanics (Nauka publishers, Moscow, 1974)

    work page 1974

  6. [6]

    E., Quantum Mechanics (World Scientific, Singapore, 2000)

    Ballentine L. E., Quantum Mechanics (World Scientific, Singapore, 2000)

    work page 2000

  7. [7]

    Kivshar Y. S. and Agrawal G. P., Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)

    work page 2003

  8. [8]

    N., Optical spa tial solitons: historical overview and recent advances, Re p

    Chen Z., Segev M., and Christodoulides D. N., Optical spa tial solitons: historical overview and recent advances, Re p. Prog. Phys. 75, 086401 (2012), doi:10.1088/0034-4885/75/8/086401

    work page doi:10.1088/0034-4885/75/8/086401 2012

  9. [9]

    D., Gillen K., and Guha S., Light Propagation in Linear Optical Media (Taylor & Francis, Boca Raton, 2014)

    Gillen G. D., Gillen K., and Guha S., Light Propagation in Linear Optical Media (Taylor & Francis, Boca Raton, 2014)

    work page 2014

  10. [10]

    S., The propagation of Hermite-Gauss wave pac kets in optics and quantum mechanics, Natural Sciences 4, e202300912 (2024), DOI: 10.1002/ntls.20230012

    Briggs J. S., The propagation of Hermite-Gauss wave pac kets in optics and quantum mechanics, Natural Sciences 4, e202300912 (2024), DOI: 10.1002/ntls.20230012

    work page doi:10.1002/ntls.20230012 2024

  11. [11]

    Wang K. L. and Wei C. F., Novel optical soliton solutions to nonlinear paraxial wave model, Mod. Phys. Lett. B 39, 2450469 (2025), https://doi.org/10.1142/S021798492450 4694

    work page doi:10.1142/s021798492450 2025

  12. [12]

    Yang J., Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, Philadelphia, 2010)

    work page 2010

  13. [13]

    J., and Kev rekidis, P

    Carretero-Gonz´ alez, R., Frantzeskakis D. J., and Kev rekidis, P. G., Nonlinear Waves & Hamiltonian Systems: From One To Many Degrees of Freedom, From Discrete To Continuum (Oxford University Press, Oxford, 2024)

    work page 2024

  14. [14]

    and Wigner E., ¨Uber merkw¨ urdige diskrete Eigenwerte, Phys

    von Neumann J. and Wigner E., ¨Uber merkw¨ urdige diskrete Eigenwerte, Phys. Z. 30, 465-467 (1929)

    work page 1929

  15. [15]

    Stillinger F. H., Potentials supporting positive-ene rgy eigenstates and their application to semiconductor het erostructures, Physica B 85, 270–276 (1976); https://doi.org/10.1016/0378-4363(76 )90021-8

    work page doi:10.1016/0378-4363(76 1976

  16. [16]

    and Wintgen, D., Interfering resonances a nd bound states in the continuum, Phys

    Friedrich H. and Wintgen, D., Interfering resonances a nd bound states in the continuum, Phys. Rev. A 32, 3231–3242 (1985); DOI: https://doi.org/10.1103/PhysRevA.32.3231

    work page doi:10.1103/physreva.32.3231 1985

  17. [18]

    C. W. Hsu, Zhen B., Stone A. D., Joannopoulos J. D., and So ljaˇ ci´ c M., Bound states in the continuum, Nature Rev. Mat. 9, 16048 (2016); https://doi.org/10.1103/PhysRevLett.10 0.183902

    work page doi:10.1103/physrevlett.10 2016

  18. [19]

    Kodigala A., Lepetit T., Gu Q., Bahari B., Fainman Y., an d Kant´ e B., Lasing action from photonic bound states in continuum, Nature 541, 196-199 (2017); https://doi.org/10.1038/nature20799

    work page doi:10.1038/nature20799 2017

  19. [20]

    L., Geng G., Zeng X., Zeng Y., Hu G., Overvi g A., Li J., Qiu C.-W., Al` u A., Kivshar Y

    Shi T., Deng Z. L., Geng G., Zeng X., Zeng Y., Hu G., Overvi g A., Li J., Qiu C.-W., Al` u A., Kivshar Y. S., and Li X., Planar chiral metasurfaces with maximal and tunable chi roptical response driven by bound states in the continuum, Nature Communications 13, 4111 (2022), https://doi.org/10.1038/s41467-022-3187 7-1

    work page doi:10.1038/s41467-022-3187 2022

  20. [21]

    Gennaro S

    Santiago-Cruz T., Sylvain D. Gennaro S. D., Mitrofanov O., Addamane S., Reno J., Brener I., and V. Chekhova M. V., Resonant metasurfaces for generating complex quantum stat es, Science 377, 991-995 (2022); DOI: 10.1126/science.abq8684

    work page doi:10.1126/science.abq8684 2022

  21. [22]

    Acharyaet al.[ALICE], Nature588, 232-238 (2020) [erratum: Nature590, E13 (2021)] doi:10.1038/s41586- 020-3001-6 [arXiv:2005.11495 [nucl-ex]]

    Chen Y., Deng H., Sha X., Chen W., Wang R., Chen Y.-H., Wu D ., Chu J.,, Kivshar Y. S., Xiao S., and Qiu C.-W., Observation of intrinsic chiral bound states in the continu um. Nature 613, 474-479 (2023); https://doi.org/10.1038/s41586- 022-05467-6

    work page doi:10.1038/s41586- 2023

  22. [23]

    T., and Xiao, M., Applications of bound states in the continuum in photonics, Nature Rev

    Kang M., Liu T., Chan C. T., and Xiao, M., Applications of bound states in the continuum in photonics, Nature Rev. Phys. 5, 659-678 (2023); https://doi.org/10.1038/s42254-023-0 0642-8

    work page doi:10.1038/s42254-023-0 2023

  23. [24]

    Powell D

    Huang L, Xu L. Powell D. A., Padilla W. J., and Miroshnich enko A. E., Resonant leaky modes in all-dielectric metasyst ems: Fundamentals and applications, Phys. Rep. 1008, 1-66 (2023 ), https://doi.org/10.1016/j.physrep.2023.01.001. 12

    work page doi:10.1016/j.physrep.2023.01.001 2023

  24. [25]

    Liu B., Peng Y., Jin Z., Wu X., Gu H., Wei. D., Zhu Y., and Zh uang S., Terahertz ultrasensitive biosensor based on wide-area and intense light-matter interaction supported by QBIC, Chemical Engineering Journal 462, 142347 (2023), https://doi.org/10.1016/j.cej.2023.142347

    work page doi:10.1016/j.cej.2023.142347 2023

  25. [26]

    Exciton polari ton condensation from bound states in the continuum at room temperature

    Wu X., Zhang S., Song J., Deng X., Du W., Zeng X., Zhang Y., Zhang Z., Chen Y., Wang Y., Jiang C., Zhong Y., Wu B., Zhu Z., Liang Y., Zhang Q., Xiong Q., and Liu X. Exciton polari ton condensation from bound states in the continuum at room temperature. Nature Communications 15, 3345 (2024); DOI10.1038/s41467-024-47669-8

    work page 2024

  26. [27]

    K., Wang S

    Zhao, W. K., Wang S. Y.,, Jing Y. H., Ge H., Wang Q., Zeng Y. Q., Jia B. W., and Xu N. Giant struc- turally modulated intrinsic chirality of quasi-bound stat es in continuum. Optics Letters 50, 1649-1652 (2025); https://doi.org/10.1364/OL.549705

    work page doi:10.1364/ol.549705 2025

  27. [28]

    Pitaevskii L. P. and Stringari S., Bose-Einstein Condensation (Oxford University Press, Oxford, 2003)

    work page 2003

  28. [29]

    E mbedded solitons

    Champneys A. R., Malomed B. A., Yang J., and Kaup D. J., “E mbedded solitons” : solitary waves in resonance with the linear spectrum, Physica D 152-153, 340-354 (2001); https://doi.org/10.1016/S0167-2789(0 1)00178-6

    work page doi:10.1016/s0167-2789(0 2001

  29. [30]

    A., Large mode-2 inte rnal solitary waves in three-layer flows, J

    Doak A., Barros R., and Milewski P. A., Large mode-2 inte rnal solitary waves in three-layer flows, J. Fluid Mech. 953, A42 (2022); doi:10.1017/jfm.2022.974

    work page doi:10.1017/jfm.2022.974 2022

  30. [31]

    A., Mathematical model with unrestricte d dispersion and polynomial nonlinearity, Appl

    Kudryashov N. A., Mathematical model with unrestricte d dispersion and polynomial nonlinearity, Appl. Math. Lett . 138, 108519 (2023), https://doi.org/10.1016/j.aml.2022.108 519

    work page doi:10.1016/j.aml.2022.108 2023

  31. [32]

    Li Z. and Peng C., Dynamics and Embedded Solitons of Stoc hastic Quadratic and Cubic Nonlinear Susceptibilities wit h Multiplicative White Noise in the Itˆ o Sense, Mathematics 11, 3185 (2023), https://doi.org/10.3390/math11143185

    work page doi:10.3390/math11143185 2023

  32. [33]

    Rizvi S. T. R., Seadawy A. R., Nimra, Ali K., and Aziz N., V ariety of optical soliton solutions via sub-ODE approach to embedded soliton generating model in quadratic nonlinear m edia, Int. J. Mod. Phys. B 37, 2350137 (2023); https://doi- org.rproxy.tau.ac.il/10.1142/S0217979223501370

    work page doi:10.1142/s0217979223501370 2023

  33. [34]

    P., Gao C., and Lin J

    Fang P. P., Gao C., and Lin J. Bifurcations and dynamics o f nonlinear excitations in twisted-bilayer optical lattic es. Chaos, Solitons & Fractals 195, 116314 (2025); DOI10.1016/j.chao s.2025.116314

    work page arXiv 2025

  34. [35]

    P., and O’Dell D

    Sundaram S., Burgess C. P., and O’Dell D. H. J., Duality b etween the quantum inverted harmonic oscillator and invers e square potentials, New J. Phys. 26, 053023 (2024); https://doi.org/10.1088/1367-2630/ad3 a91

    work page doi:10.1088/1367-2630/ad3 2024

  35. [36]

    and Malomed B

    Hacker N. and Malomed B. A., Trapping wave fields in an exp ulsive potential by means of linear coupling, Phys. Rev. E 105, 034213 (2022); DOI: https://doi.org/10.1103/PhysRe vE.105.034213

    work page doi:10.1103/physre 2022

  36. [37]

    Reyna A. S. and de Ara´ ujo C. B., High-order optical nonl inearities in plasmonic nanocomposites – a review, Adv. Opt . Phot. 9, 720-774 (2017); DOI:10.1364/AOP.9.000720

    work page doi:10.1364/aop.9.000720 2017

  37. [38]

    and Michinel H., Stable azimuthal stationary state in quintic nonlinear optical media, J

    Quiroga-Teixeiro M. and Michinel H., Stable azimuthal stationary state in quintic nonlinear optical media, J. Opt . Soc. Am. B 14, 2004-2009 (1997); https://doi.org/10.1364/JOSAB.14.0 02004

    work page doi:10.1364/josab.14.0 2004

  38. [40]

    Bakkali-Hassani B., Maury C., Stringari S., Nascimben e S., Dalibard J., and Beugnon J., The cross-over from Townes solitons to droplets in a 2D Bose mixture, New J. Phys. 25, 013007 (2023); https://doi.org/10.1088/1367-2630/aca ee3

    work page doi:10.1088/1367-2630/aca 2023

  39. [41]

    I., Walther V., and Hung C.-L.; Coll apse of a Quantum Vortex in an Attractive Two-Dimensional Bose Ga s, Phys

    Banerjee S., Zhou K., Kant Tiwari S., Tamura, Li R., Kevr ekidis P., Mistakidis S. I., Walther V., and Hung C.-L.; Coll apse of a Quantum Vortex in an Attractive Two-Dimensional Bose Ga s, Phys. Rev. Lett. 135, 073401 (2025); DOI:10.1103/c6wx- zc9x

    work page doi:10.1103/c6wx- 2025

  40. [42]

    Mansuryan T., Bagley N., Boulesteix R., Arosa Y., Krupa K., Wetzel B., Chaves B. P., Wabnitz S., Couderc V., Aceves A., and Tonello A., Nonlinear flying focus pulses for ultrafa st 3D nonlinear microscopy, Optica 12, 1192-1199 (2025); https://doi.org/10.1364/OPTICA.561661

    work page doi:10.1364/optica.561661 2025

  41. [43]

    M., Smirne G., Cassettari D., Chandra shekar, C

    Boyer V., Godun R. M., Smirne G., Cassettari D., Chandra shekar, C. M., Deb A. B., Laczik Z. J., and Foot C. J., Dynamic manipulation of Bose-Einstein condensates with a s patial light modulator, Phys. Rev. A 73, 031402 (2006); DOI: https://doi.org/10.1103/PhysRevA.73.031402

    work page doi:10.1103/physreva.73.031402 2006

  42. [44]

    J., Perry A

    Lin Y. J., Perry A. R., Compton, R. L., Spielman, I. B., an d Porto J. V., Rapid production of 87Rb Bose- Einstein condensates in a combined magnetic and optical pot ential, Phys. Rev. A 79, 063631 (2009); DOI: https://doi.org/10.1103/PhysRevA.79.063631

    work page doi:10.1103/physreva.79.063631 2009

  43. [45]

    He C., Shen Y., and Forbes A., Towards higher-dimension al structured light, Light: Science & Applications 11, 205 (2022); https://doi.org/10.1038/s41377-022-00897-3 www.natur e.com/lsa

    work page doi:10.1038/s41377-022-00897-3 2022

  44. [46]

    Bai Y., Lv H., Fu X., and Yang Y., Vortex beam: generation and detection of orbital angular momentum [Invited], Chine se Optics Letters 20, 012601 (2022); DOI: 10.3788/COL202220.012601

    work page doi:10.3788/col202220.012601 2022

  45. [47]

    Ahmed H., Kim H., Zhang Y., Intaravanne Y., Jang J., Rho J ., Chen S., and Chen X., Optical metasurfaces for gener- ating and manipulating optical vortex beams, Nanophotonic s 11, 941-956 (2022); https//doi.org/10.1515/nanoph-2021- 0746Received November 29, 2021; accepted D

    work page doi:10.1515/nanoph-2021- 2022

  46. [48]

    Lian Y., Qi X., Wang Y., Bai Z., Wang Y., and Lu Z., OAM beam generation in space and its applications: A review, Optics and Lasers in Engineering 151, 106923 (2022); https://doi.org/10.1016/j.optlaseng.2 021.106923

    work page doi:10.1016/j.optlaseng.2 2022

  47. [49]

    P., and Hadzibabic Z., Quantum gases i n optical boxes, Nature Physics 17, 1334-1341 (2021); DOI10.1038/s41567-021-01403-z

    Navon N., Smith R. P., and Hadzibabic Z., Quantum gases i n optical boxes, Nature Physics 17, 1334-1341 (2021); DOI10.1038/s41567-021-01403-z

    work page 2021

  48. [50]

    Lee K., Kim J., and Shin Y., Flattening a trapped a tomic gas using a programmable optical potential in a feedba ck loop, Phys

    Kim, S. Lee K., Kim J., and Shin Y., Flattening a trapped a tomic gas using a programmable optical potential in a feedba ck loop, Phys. Rev. A 110 (2024); DOI: 10.1103/PhysRevA.110.033326

    work page doi:10.1103/physreva.110.033326 2024

  49. [51]

    F., Huh S

    Impertro A., Karch S., Wienand J. F., Huh S. J., Schweize r C., Bloch I., and Aidelsburger M., Local Readout and Control of Current and Kinetic Energy Operators in Optical L attices, Phys. Rev. Lett. 133, 063401 (2024); DOI: https://doi.org/10.1103/PhysRevLett.133.063401. 13

    work page doi:10.1103/physrevlett.133.063401 2024

  50. [52]

    Bashan G., Eyal A., Tur M., and Arie A., Optically progra mmable quasi phase matching in four-wave mixing, Nature Communications 16, 6855 (2025); DOI10.1038/s41467-025-62025-0

    work page 2025

  51. [53]

    Olshanii M., Perrin H., and Lorent V., Example of a Quant um Anomaly in the Physics of Ultracold Gases, Phys. Rev. Lett. 105, 095302 (2010); DOI: https://doi.org/10.1103/PhysRevLe tt.105.095302

    work page doi:10.1103/physrevle 2010

  52. [55]

    I, Exact solutions to fall of particle to si ngular potential: classical versus quantum cases, Proc

    Tribelsky M. I, Exact solutions to fall of particle to si ngular potential: classical versus quantum cases, Proc. R. Soc. A 479, 20230366 (2023); https://doi.org/10.1098/rspa.2023.0 366

    work page doi:10.1098/rspa.2023.0 2023

  53. [56]

    Zang Y., Gu Y., and Jiang S., Detecting Quantum Anomalie s in Open Systems, Phys. Rev. Lett. 133, 106503 (2024); DOI: https://doi.org/10.1103/PhysRevLett.133.106503

    work page doi:10.1103/physrevlett.133.106503 2024

  54. [57]

    Alexander T. J. and Berg´ e L., Ground states and vortice s of matter-wave condensates and optical guided waves, Phys . Rev. E 65, 026611 (2002); DOI: https://doi.org/10.1103/PhysRevE. 65.026611

    work page doi:10.1103/physreve 2002

  55. [58]

    Carr L. D. and Clark C. W., Vortices in attractive Bose-E instein condensates in two dimensions, Phys. Rev. Lett. 97, 010403 (2006); DOI: https://doi.org/10.1103/PhysRevLet t.97.010403

    work page doi:10.1103/physrevlet 2006

  56. [59]

    arXiv:1907.07913, doi:10.1103/PhysRevA

    Mihalache D., Mazilu D., Malomed B. A., and Lederer F., V ortex stability in nearly-two-dimensional Bose-Einstein con- densates with attraction,. Phys. Rev. A 73, 043615 (2006); DOI: https://doi.org/10.1103/PhysRevA. 73.043615

    work page doi:10.1103/physreva 2006

  57. [60]

    N., On the theory of superfluidity, Izv

    Bogolyubov, N. N., On the theory of superfluidity, Izv. A kad. Nauk SSSR, Ser. Fiz. 11, 77-90 (1947)

    work page 1947

  58. [61]

    and Promislow K., Spectral and Dynamical Stability of Nonlinear Waves , Springer-Verlag (New York, 2013)

    Kapitula T. and Promislow K., Spectral and Dynamical Stability of Nonlinear Waves , Springer-Verlag (New York, 2013)

    work page 2013

  59. [62]

    Sadaka G., Jolivet P., Charalampidis E.G., and Danaila I., Parallel finite-element codes for the Bogoliubov- de Gennes stability analysis of Bose-Einstein condensates , Comput. Phys. Commun. 306, 109378 (2025); https://doi.org/10.1016/j.cpc.2024.109378

    work page doi:10.1016/j.cpc.2024.109378 2025