Pith. sign in

REVIEW 1 major objections 5 minor 53 references

Dimer problem on a spherical surface

T0 review · 1 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A dimer on a sphere turns quasi-one-dimensional when it rotates fast, with an energy that depends on total angular momentum.

desk verdict Solid two-body result for dimers on a sphere with genuine j-dependence; the only real soft spot is the quasi-1D crossover extrapolation, which is probably right but should be checked. read the letter →

arxiv 2502.06724 v2 pith:KXOPNQOI submitted 2025-02-10 cond-mat.quant-gas quant-ph

A. Tononi , D. S. Petrov , M. Lewenstein This is my paper · ORCID
classification cond-mat.quant-gasquant-ph
keywords two-bodyproblemonaspheredimertotalangularmomentumquasi-one-dimensionalshell-shapedquantumgaseszero-rangeinteractionsultracoldatoms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper solves two atoms on a sphere interacting through a zero-range s-wave potential, for a fixed total angular momentum $j$. It shows that the binding energy and the relative wave function depend on $j$: at large $j$, the centrifugal motion acts like an effective harmonic trap perpendicular to the direction of motion, squeezing the dimer from a two-dimensional shape into a quasi-one-dimensional one. The authors derive a closed relation, Eq. (8), connecting the scattering length $a$ to the dimer energy $E = j^2/4 + j/2 + q^2$ in that regime, and map out three regimes: a compact isotropic dimer, a quasi-one-dimensional dimer, and a pair delocalized along the equator. This matters because shell-shaped ultracold gases can be rotated, and the two-body spectrum is the starting point for their many-body behavior.

What carries the argument

The machinery is a rigid-rotor decomposition: the two-body kinetic energy is written in a body-fixed frame attached to the dimer, separating rotational motion of the molecular axis from relative motion along the geodesic angle $\theta$. Expanding the wave function in Wigner-D functions $D^j_{ml}$ turns the Schrödinger equation into a finite set of coupled ordinary differential equations in $\theta$, Eq. (5), with the s-wave interaction entering through the Bethe-Peierls boundary condition. The large-$j$ step then replaces the centrifugal potential near the equator by a harmonic oscillator of frequency $j/2$, producing an effective flat-space quasi-one-dimensional problem whose solution is built from a harmonic-oscillator Green function and leads to Eq. (8).

What would settle it

Solve the full coupled equations (5) numerically for intermediate and large $j$ (for example $j=20$ and $j=30$) across the window $1/\sqrt{j} < a < a_*$ and compare the exact energy with Eq. (8); if the match degrades as $q^2$ approaches $-1$ instead of holding within $O(1/j)$, the assumed crossover boundary $a_*$ is incorrect.

Watch

Extended reading notes

Core claim

On a sphere, the center-of-mass and relative motions of a dimer do not separate, so the dimer's binding energy and wave function depend on the total angular momentum $j$. For large $j$, the centrifugal potential near the equator acts as a harmonic confinement with frequency $j/2$ and oscillator length $\sim R/\sqrt{j}$, reducing the two-body problem to a flat-space quasi-one-dimensional dimer in that trap. The resulting energy is $E = j^2/4 + j/2 + q^2$, where $q^2$ is fixed by Eq. (8) in terms of the scattering length $a$; the relation identifies the quasi-one-dimensional window $R/\sqrt{j} \lesssim a \lesssim a_*$ with $a_* \approx e^{\sqrt{\pi j/2}}R$, beyond which the pair delocalizes along the equator while staying localized near it with polar-angle spread $\sim 1/\sqrt{j}$. For $j=0$ and $j=1$ the paper gives exact solutions in terms of Legendre and Jacobi functions, and for $j>1$ it solves the coupled equations numerically.

Load-bearing premise

The central assumption is that the large-angular-momentum reduction to a harmonic trap perpendicular to the equator stays accurate all the way to weakly bound states at the edge of the quasi-one-dimensional regime, although the formal derivation only guarantees it for stronger binding.

Editorial extensions

If this is right

  • For large $j$, the dimer energy is $E = j^2/4 + j/2 + q^2$, with the $j^2/4$ term from center-of-mass motion along the equator and the $j/2$ terms from zero-point energy in the perpendicular harmonic confinement.
  • The anisotropy appears already at $j=2$ and becomes pronounced at large $j$: in the quasi-1D regime the dimer's transverse size is $\sim 1/\sqrt{j}$ while its length along the motion is $1/\sqrt{-q^2}$.
  • At the crossover $a_* \approx e^{\sqrt{\pi j/2}}$, the dimer becomes delocalized along the equator but remains localized in the polar direction with spread $\sim 1/\sqrt{j}$; this crossover would be observable as a sharp change in binding energy as $a$ is varied.
  • The two-body spectrum can be probed by radio-frequency spectroscopy, and the anisotropic shape should show up in time-of-flight expansion of shell-shaped ultracold gases.
  • For small $j$, increasing $a$ simply enlarges an isotropic dimer until it reaches the sphere radius, so the squeezing effect is specific to finite angular momentum.

Reading between the lines

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

  • The same centrifugal-squeezing mechanism should operate in any curved shell with a conserved angular momentum about an axis, so a rotating slightly elliptical bubble would produce the same effective one-dimensional confinement with a modified oscillator length.
  • The quasi-one-dimensional enhancement of binding at finite $j$ implies that, in a rotating Fermi gas on a shell, the BCS-BEC crossover boundary should shift with rotation frequency, making pair formation easier at fixed scattering length.
  • An experimental test could use RF association spectroscopy on a phase-imprinted, rapidly rotating shell-shaped gas: the dimer resonance should split with $j$, since the binding energy at fixed $a$ changes with total angular momentum.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper studies two particles with zero-range s-wave interactions confined to the surface of a sphere. Using a rigid-rotor parametrization and the Laplace-Beltrami operator, the authors derive a set of coupled ordinary differential equations (Eq. (5)) for the relative wave function at fixed total angular momentum j. They solve the j=0 and j=1 cases analytically, solve j>1 numerically, and construct a large-j quasi-one-dimensional reduction in which the dimer moves along the equator under a harmonic confinement with oscillator length ~1/sqrt(j). The central result is that the dimer's binding energy and wave function depend strongly on j: the molecule becomes squeezed perpendicular to the center-of-mass motion and enters a quasi-1D regime for scattering lengths 1/sqrt(j) less than or similar to a less than or similar to a*, with a* approximately exp(sqrt(pi j/2)). The authors argue that this curvature- and angular-momentum-induced dimensional crossover is relevant to ultracold shell-shaped gases.

Significance. If the results are correct, the paper provides the first solution of the two-body problem on a sphere with finite total angular momentum, showing a nontrivial coupling between relative and center-of-mass motion in a curved geometry. The derivation is internally consistent: Eq. (5) follows from the metric, the j=0 and j=1 limits are exact, and the small-a and large-a asymptotes match the numerical curves. The quasi-1D reduction is a useful conceptual tool, and the predicted crossover scattering length a* is a concrete, falsifiable prediction for experiments with shell-shaped gases. The paper is clearly written and the supplemental material contains the metric and the derivation of the quasi-1D equation. The main weakness is the unproved extrapolation of Eq. (8) to q^2=-1, which defines a*; this does not affect the qualitative claim but leaves a quantitative boundary insufficiently supported.

major comments (1)
  1. [Eq. (8) and Supplemental Material Appendix C] The quantitative boundaries of the quasi-1D regime rely on using Eq. (8) outside the domain stated for its derivation. The paper requires 1 << -q^2 <~ j for Eq. (8), but a* is set by q^2 = -1; and for a -> infinity the same equation is continued to positive q^2 to reproduce the noninteracting limits. The Supplement justifies the q^2 ~ 1 extension by asserting that first-order and higher-order energy shifts relative to the harmonic-oscillator problem are of order max{q^2,1}/j, but no derivation of this assertion is given. Because the location of a* and the shape of the quasi-1D window in Fig. 2 are explicit quantitative claims, please either prove the perturbation bound or test the extrapolated Eq. (8) directly against numerical solutions of the exact coupled equations (5) for moderate j (e.g., j = 8, 10, 12).
minor comments (5)
  1. [j=1 paragraph] The line 'with nu = E1/2 1' is garbled; please provide a clean definition of nu (e.g., nu = sqrt(E1)/2).
  2. [Fig. 2 caption] The caption mentions 'thick dashed lines' for the quasi-1D theory while the text refers to 'dashed curves'; unify the notation.
  3. [References [32] and [45]] The journal name 'A VS Quantum Sci.' should be 'AVS Quantum Sci.'.
  4. [Eq. (4)] The summation should be typeset in standard form (e.g., sum over l=0, l even, up to j) rather than 'jX l=0, l even'.
  5. [Section 'We can now summarize...'] The aspect ratio used to define the left border of the quasi-1D window (a ~ 1/sqrt(j)) is not defined; please give its expression (e.g., longitudinal size over transverse size).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the j-dependent dimer spectrum is derived from the spherical metric and Bethe-Peierls boundary condition without fitting to the target result.

full rationale

The paper derives the two-body kinetic energy operator from the Laplace-Beltrami operator in the co-moving chart (Supplemental Material B), then reduces the Schr\"odinger equation to the coupled-channel system (5) by angular-momentum algebra. The large-j quasi-1D reduction (Supplemental Material C) is an approximation obtained by expanding the single-particle kinetic terms about the equator and verifying the localization scale a posteriori; the energy relation Eq. (8) is a derived Bethe-Peierls condition, not a fitted expression. The self-citations [31,32] provide the j=0 solution and scattering inputs, but these are not equivalent to the central claim about finite-j squeezing and quasi-1D geometry, which is established by the paper's own numerical solution of Eqs. (5) and the analytic solution of Eq. (8). The extrapolation of Eq. (8) to q^2=-1 to define a* is outside the formal domain stated in the main text, but this is a correctness/approximation concern, not circularity; the Supplemental Material gives an order-of-magnitude estimate for the omitted periodic-image term. No equation is defined in terms of the result it is supposed to predict, and no fitted parameter is renamed as a prediction.

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

The model has no free fitted parameters: the scattering length a is an input, and all effective potentials (e.g., the harmonic confinement at large j) are derived from the metric and the j expansion. The main axioms are the thin-shell reduction, the Bethe-Peierls zero-range condition, the even-l truncation, and the large-j harmonic approximation.

assumptions (4)
  • domain assumption Thin-shell approximation: radial motion is frozen and the system is exactly 2D on a sphere of radius R.
    Explicitly assumed in main text after Eq. (7): 'we assume the thin-shell regime completely neglecting the degree of freedom perpendicular to the sphere surface.'
  • domain assumption Zero-range s-wave interactions are described by the Bethe-Peierls boundary condition Psi|theta->0 proportional to ln(theta/a).
    Standard for ultracold atoms; used throughout to relate energy to scattering length.
  • domain assumption For large j, the j^2 and j terms in Eq. (S6) suffice; higher-order terms are perturbations.
    The harmonic confinement expansion in Appendix C truncates the kinetic operator at order j.
  • domain assumption Only even-l channels are coupled by the s-wave interaction; odd-l channels are noninteracting.
    Main text: 'the s-wave interaction is effective only in the equation with l = 0 because the other components experience the centrifugal barrier l^2 B(theta) proportional to 1/theta^2.'

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dimer problem on a spherical surface." pith.science (2026). https://pith.science/paper/KXOPNQOI

@misc{pith2026250206724,
  author       = {Pith},
  title        = {Pith review of: Dimer problem on a spherical surface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXOPNQOI}},
  note         = {Machine review of arXiv:2502.06724}
}
read the original abstract

We solve the problem of a dimer moving on a spherical surface and find that its binding energy and wave function are sensitive to the total angular momentum. The dimer gets squeezed in the direction orthogonal to the center-of-mass motion and can qualitatively change its geometry from two-dimensional to one-dimensional. These results suggest that combining the curved geometry with finite angular momentum may give rise to qualitatively new many-body phenomena in ultracold shell-shaped gases.

Figures

Figures reproduced from arXiv: 2502.06724 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the coordinate system of two particles [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dimer energy spectrum versus [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Contour plots of the ratio (rescaled to its max [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 46 canonical work pages

  1. [1]

    Einstein, L

    A. Einstein, L. Infeld, and B. Hoffmann, The Gravita- tional Equations and the Problem of Motion, Annals of Mathematics 39, 65 (1938)

    work page 1938

  2. [2]

    European Union NextGenerationEU/PRTR

    The dimer in these cases is isotropic although the θ dependence of its wave func- tion is sensitive to j. The anisotropy first appears in the case j = 2 where ψ2 ̸= 0. It manifests itself in a squeezing of the molecule along a direction which de- pends on the center-of-mass angles α and β and on m (note, however, that ψl depend on j, but not on m). The ph...

    arXiv 2022

  3. [3]

    B. M. Barker and R. F. O’Connell, Gravitational two-body problem with arbitrary masses, spins, and quadrupole moments, Phys. Rev. D 12, 329 (1975)

    work page 1975

  4. [4]

    Buonanno and T

    A. Buonanno and T. Damour, Effective one-body ap- proach to general relativistic two-body dynamics, Phys. Rev. D 59, 084006 (1999)

    work page 1999

  5. [5]

    Schild, Electromagnetic Two-Body Problem, Phys

    A. Schild, Electromagnetic Two-Body Problem, Phys. Rev. 131, 2762 (1963)

    work page 1963

  6. [6]

    Landau and E.M

    L.D. Landau and E.M. Lifshitz, Quantum Mechanics, (Butterworth-Heinemann, Oxford 1981)

    work page 1981

  7. [7]

    Dalfovo, S

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999)

    1999

  8. [8]

    Giorgini, L

    S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys. 80, 1215 (2008)

    2008

Show all 53 references
  1. [9]

    Olshanii, Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons, Phys

    M. Olshanii, Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons, Phys. Rev. Lett. 81, 938 (1998)

    1998

  2. [10]

    D. S. Petrov and G. V. Shlyapnikov, Interatomic colli- sions in a tightly confined Bose gas, Phys. Rev. A 64, 012706 (2001)

    2001

  3. [11]

    Busch, B

    T. Busch, B. G. Englert, K. Rza˙ zewski, and M. Wilkens, Two cold atoms in a harmonic trap, Found. Phys. 28, 549 (1998)

    1998

  4. [12]

    St¨ oferle, H

    T. St¨ oferle, H. Moritz, K. G¨ unter, M. K¨ ohl, and T. Esslinger, Molecules of Fermionic Atoms in an Optical Lattice, Phys. Rev. Lett. 96, 030401 (2006)

    2006

  5. [13]

    Thalhammer, K

    G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J. Hecker Denschlag, Long-Lived Feshbach Molecules in a Three-Dimensional Optical Lattice, Phys. Rev. Lett. 96, 050402 (2006)

    2006

  6. [14]

    Zwerger, ed., The BCS-BEC crossover and the uni- tary Fermi gas , Vol

    W. Zwerger, ed., The BCS-BEC crossover and the uni- tary Fermi gas , Vol. 836 (Springer, 2011)

    2011

  7. [15]

    C.-L. Hung, X. Zhang, N. Gemelke, and C. Chin, Ob- servation of scale invariance and universality in two- dimensional Bose gases, Nature 470, 236 (2011)

    2011

  8. [16]

    Yefsah, R

    T. Yefsah, R. Desbuquois, L. Chomaz, K. J. G¨ unter, and J. Dalibard, Exploring the Thermodynamics of a Two- Dimensional Bose Gas, Phys. Rev. Lett. 107, 130401 (2011)

    2011

  9. [17]

    Bakkali-Hassani, C

    B. Bakkali-Hassani, C. Maury, Y.-Q. Zou, ´E. Le Cerf, R. Saint-Jalm, P.C.M. Castilho, S. Nascimbene, J. Dalibard, and J. Beugnon, Realization of a Townes Soliton in a Two-Component Planar Bose Gas, Phys. Rev. Lett. 127, 023603 (2021)

    2021

  10. [18]

    Chen and C.-L

    C.-A. Chen and C.-L. Hung, Observation of Scale Invari- ance in Two-Dimensional Matter-Wave Townes Solitons, Phys. Rev. Lett. 127, 023604 (2021)

    2021

  11. [19]

    P. O. Fedichev, M. J. Bijlsma, and P. Zoller, Extended Molecules and Geometric Scattering Resonances in Op- tical Lattices, Phys. Rev. Lett. 92, 080401 (2004)

    2004

  12. [20]

    Schneider, S

    P.-I. Schneider, S. Grishkevich, and A. Saenz, Ab ini- tio determination of Bose-Hubbard parameters for two ultracold atoms in an optical lattice using a three-well potential, Phys. Rev. A 80, 013404 (2009)

    2009

  13. [21]

    Massignan and Y

    P. Massignan and Y. Castin, Three-dimensional strong localization of matter waves by scattering from atoms in a lattice with a confinement-induced resonance, Phys. Rev. A 74, 013616 (2006)

    2006

  14. [22]

    Nishida and S

    Y. Nishida and S. Tan, Universal Fermi Gases in Mixed Dimensions, Phys. Rev. Lett. 101, 170401 (2008)

    2008

  15. [23]

    Lamporesi, J

    G. Lamporesi, J. Catani, G. Barontini, Y. Nishida, M. Inguscio, and F. Minardi, Scattering in Mixed Dimen- sions with Ultracold Gases, Phys. Rev. Lett. 104, 153202 (2010)

    2010

  16. [24]

    D. Xiao, R. Zhang, and P. Zhang, Confinement Induced Resonance with Weak Bare Interaction in a Quasi 3+ 0 Dimensional Ultracold Gas, Few-Body Systems 60, 63 (2019)

    2019

  17. [25]

    E. L. Bolda, E. Tiesinga, and P. S. Julienne, Ultracold dimer association induced by a far-off-resonance optical lattice, Phys. Rev. A 71, 033404 (2005)

    2005

  18. [26]

    Sala and A

    S. Sala and A. Saenz, Theory of inelastic confinement- induced resonances due to the coupling of center-of-mass and relative motion, Phys. Rev. A 94, 022713 (2016)

    2016

  19. [27]

    Peano, M

    V. Peano, M. Thorwart, C. Mora, and R. Egger, Confinement-induced resonances for a two-component ultracold atom gas in arbitrary quasi-one-dimensional 6 traps, New J. Phys. 7, 192 (2005)

    2005

  20. [28]

    Melezhik and P

    V. Melezhik and P. Schmelcher, Quantum dynamics of resonant molecule formation in waveguides, New J. Phys. 11, 073031 (2009)

    2009

  21. [29]

    Tononi and L

    A. Tononi and L. Salasnich, Low-dimensional quantum gases in curved geometries, Nat. Rev. Phys. 5, 398 (2023)

    2023

  22. [30]

    Shi and H

    Z.-Y. Shi and H. Zhai, Emergent gauge field for a chiral bound state on curved surface, J. Phys. B 50, 184006, (2017)

    2017

  23. [31]

    Zhang, T.-L

    J. Zhang, T.-L. Ho, Potential Scattering on a Spherical Surface, J. Phys. B 51, 115301 (2018)

    2018

  24. [32]

    Tononi, Scattering theory and equation of state of a spherical two-dimensional Bose gas, Phys

    A. Tononi, Scattering theory and equation of state of a spherical two-dimensional Bose gas, Phys. Rev. A 105, 023324 (2022)

    2022

  25. [33]

    Tononi, G

    A. Tononi, G. E. Astrakharchik, and D. S. Petrov, Gas- to-soliton transition of attractive bosons on a spherical surface, A VS Quantum Sci.6, 023201 (2024)

    2024

  26. [34]

    Ouvry and A

    S. Ouvry and A. P. Polychronakos, Anyons on the sphere: Analytic states and spectrum, Nucl. Phys. B. 949, 114797 (2019)

    2019

  27. [35]

    A. P. Polychronakos and S. Ouvry, Two anyons on the sphere: Nonlinear states and spectrum, Nucl. Phys. B. 951, 114906 (2020)

    2020

  28. [36]

    Y. Guo, R. Dubessy, M. de Go¨ er de Herve, A. Kumar, T. Badr, A. Perrin, L. Longchambon, and H. Perrin, Super- sonic Rotation of a Superfluid: A Long-Lived Dynamical Ring, Phys. Rev. Lett. 124, 025301 (2020)

    2020

  29. [37]

    R. A. Carollo, D. C. Aveline, B. Rhyno, S. Vishveshwara, C. Lannert, J. D. Murphree, E. R. Elliott, J. R. Williams, R. J. Thompson, and N. Lundblad, Observation of ultra- cold atomic bubbles in orbital microgravity, Nature 606, 281 (2022)

    2022

  30. [38]

    F. Jia, Z. Huang, L. Qiu, R. Zhou, Y. Yan, and D. Wang, Expansion Dynamics of a Shell-Shaped Bose-Einstein Condensate, Phys. Rev. Lett. 129, 243402 (2022)

    2022

  31. [39]

    Huang, K.Y

    Z. Huang, K.Y. Lee, C.K. Wong, L. Qiu, B. Yang, Y. Yan, and D. Wang, Probing the hollowing transition of a shell- shaped BEC with collective excitation, arXiv:2503.12318

    arXiv

  32. [40]

    Fernholz, R

    T. Fernholz, R. Gerritsma, P. Kr¨ uger, and R. J. C. Spreeuw, Dynamically controlled toroidal and ring- shaped magnetic traps, Phys. Rev. A 75, 063406 (2007)

    2007

  33. [41]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- sonskii, Quantum theory of angular momentum, (World scientific, 1988)

    1988

  34. [42]

    Gordy and R

    W. Gordy and R. L. Cook, Microwave molecular spectra, 3rd ed., (New York, Wiley, 1984). See chapter VII.2

    1984

  35. [43]

    Fedotova and N

    I. Fedotova and N. Virchenko, Generalized Associated Legendre Functions and Their Applications (World Sci- entific Publishing, 2001)

    2001

  36. [44]

    Originally we reported the j = 1 solution in terms of hypergeometric functions

    We thank one of the Referees for suggesting this form. Originally we reported the j = 1 solution in terms of hypergeometric functions

  37. [45]

    Moritz Carmesin and M

    C. Moritz Carmesin and M. A. Efremov, Confine- ment Induced Resonances in Spherical Shell Traps, arXiv:2401.14946

    arXiv

  38. [46]

    Beregi, C

    A. Beregi, C. Foot, and S. Sunami, Quantum simula- tions with bilayer 2D Bose gases in multiple-RF-dressed potentials featured, A VS Quantum Sci.6, 030501 (2024)

    2024

  39. [47]

    Ma and X

    Y. Ma and X. Cui, Shell-Shaped Quantum Droplet in a Three-Component Ultracold Bose Gas, Phys. Rev. Lett. 134, 043402 (2025)

    2025

  40. [48]

    Sharma, D

    R. Sharma, D. Rey, L. Longchambon, A. Perrin, H. Per- rin, and R. Dubessy, Thermal Melting of a Vortex Lattice in a Quasi-Two-Dimensional Bose Gas, Phys. Rev. Lett. 133, 143401 (2024)

    2024

  41. [49]

    Dubessy and H

    R. Dubessy and H. Perrin, Perspective: Quantum gases in bubble traps, A VS Quantum Sci. 7, 010501 (2025)

    2025

  42. [50]

    Lannert, T.-C

    C. Lannert, T.-C. Wei, and S. Vishveshwara, Dynamics of condensate shells: Collective modes and expansion, Phys. Rev. A 75, 013611 (2007)

    2007

  43. [51]

    Tononi, F

    A. Tononi, F. Cinti, and L. Salasnich, Quantum bubbles in microgravity, Phys. Rev. Lett. 125, 010402 (2020)

    2020

  44. [52]

    Y. He, H. Guo, and C.-C. Chien, BCS-BEC crossover of atomic Fermi superfluid in a spherical bubble trap, Phys. Rev. A 105, 033324 (2022)

    2022

  45. [53]

    R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). Supplemental Material: Dimer spectrum on a spherical surface A) Particle positions in⃗ ucoordinates The particle positions ⃗ r1 and ⃗ r2 can be expressed in terms of ⃗ nc and ⃗ n...

    1965

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.