\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

The mean-field limit of the Lieb-Liniger model

  • *Corresponding author: Matthew Rosenzweig

    *Corresponding author: Matthew Rosenzweig

The author acknowledges financial support from the University of Texas at Austin and the Simons Collaboration on Wave Turbulence

Abstract Full Text(HTML) Figure(0) / Table(1) Related Papers Cited by
  • We consider the well-known Lieb-Liniger (LL) model for $ N $ bosons interacting pairwise on the line via the $ \delta $ potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the $ \delta $ potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the $ N $-body wave function in a single particle variable. By further exploiting the $ L^2 $-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of $ N $-body initial states.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q40.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  Notation

    Symbol Definition
    $ A\lesssim B, \ A\sim B $ There are absolute constants $ C_1, C_2>0 $ such that $ A\leq C_1 B $ or $ C_2B\leq A\leq C_1 B $; subscript $ p $ (e.g. $ A\lesssim_p B $) denotes dependence on parameter $ p $
    $ \underline{x}_{k}, \ \underline{x}_{i;i+k} $ $ (x_1, \ldots, x_k), \ (x_i, \ldots, x_{i+k}) $, where $ x_j\in {\mathbb{R}} $ for $ j\in\left\{ {1, \ldots, k} \right\} $ or $ j\in\left\{ {i, \ldots, i+k} \right\} $
    $ d \underline{x}_{k}, \ d \underline{x}_{i;i+k} $ $ dx_1 \cdots dx_k, \ dx_i \cdots dx_{i+k} $
    $ {\mathbb{N}}, \ {\mathbb{N}}_0 $ natural numbers, natural numbers inclusive of zero
    $ \left\langle {\cdot}|{\cdot} \right\rangle $ $ L^2( {\mathbb{R}}^N) $ inner product with physicist's convention: $\left\langle {f}|{g}\right\rangle := \int_{ {\mathbb{R}}^N}d \underline{x}_N \overline{f( \underline{x}_N)}g( \underline{x}_N) $
    $ \left\langle \cdot , \cdot \right\rangle $ duality pairing
    $ \left\langle {{\cdot}} \right| \ \left| {{\cdot}} \right\rangle $ Dirac's bra-ket notation: see footnote 2
    $ A_{i_1\cdots i_k}^{(k)} $ subscript denotes that the operator on $ L^2( {\mathbb{R}}^N) $ acts in the variables $ (x_{i_1}, \ldots, x_{i_k}) $
    $ \varphi^{\otimes k} $ $ k $-fold tensor product of $ \varphi $ with itself realized as $ \varphi^{\otimes k}( \underline{x}_k) = \prod_{i=1}^k \varphi(x_i), \ \underline{x}_k\in {\mathbb{R}}^k $
    $ {\rm{Tr}}_{1, \ldots, N} $ trace on $ L^2( {\mathbb{R}}^N) $
    $ {\rm{Tr}}_{k+1, \ldots, N} $ partial trace on $ L^2( {\mathbb{R}}^N) $ over $ x_{k+1}, \ldots, x_N $ coordinates
    $ {\bf{1}}, \ {\bf{1}}_N $ identity operator on $ L^2( {\mathbb{R}}) $ and on $ L^2( {\mathbb{R}}^N) $
    $ H_{N}, \ H_{N, \varepsilon} $ LL Hamiltonian and regularized LL Hamiltonian: see (1.2) and (1.15)
    $ p_j, \ q_j $ projectors $ {\bf{1}}^{\otimes j-1}\otimes p \otimes {\bf{1}}^{N-j}, \ {\bf{1}}^{\otimes j-1} \otimes q \otimes {\bf{1}}^{N-j} $: see (3.2)
    $ P_k $ projector onto subspace of $ k $ particles not in the state $ \varphi(t) $: see (3.4)
    $ \widehat{f}, \ \widehat{f}^{-1} $ operator $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $ defined by $ \widehat{f} := \sum_{k=0}^N f(k)P_k $, for $ f: {\mathbb{Z}}\rightarrow {\mathbb{C}} $: see (3.6)
    $ n_N, m_N $ $ \widehat{n_N}, \widehat{m_N} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see 3.1
    $ \mu, \nu $ $ \widehat{\mu}, \widehat{\nu} $ functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see (3.33) and (3.52)
    $ \beta_N $ time-dependent functional of solution $ \varphi $ to (1.8) and $ \Phi_N $ to (1.3): see 3.1
    $ \tau_n $ shift operator on $ {\mathbb{C}}^{ {\mathbb{Z}}} $: see (3.16)
    $ \underline{\Delta}_k $ Laplacian on $ {\mathbb{R}}^k $: $ \underline{\Delta}_k := \sum_{i=1}^k \Delta_i $
    $ \left[ \cdot , \cdot \right]$ commutator bracket: $\left[ A, B \right] := AB-BA $
     | Show Table
    DownLoad: CSV
  • [1] R. AdamiC. BardosF. Golse and A. Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal., 40 (2004), 93-108. 
    [2] R. AdamiF. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127 (2007), 1193-1220.  doi: 10.1007/s10955-006-9271-z.
    [3] Z. Ammari and S. Breteaux, Propagation of chaos for many-boson systems in one dimension with a point pair-interaction, Asymptot. Anal., 76 (2012), 123-170.  doi: 10.3233/ASY-2011-1064.
    [4] C. BardosF. Golse and N. J. Mauser, Weak coupling limit of the $N$-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), 275-293.  doi: 10.4310/MAA.2000.v7.n2.a2.
    [5] N. Benedikter, M. Porta and B. Schlein, Effective Evolution Equations from Quantum Dynamics, SpringerBriefs in Mathematical Physics, 7. Springer, Cham, 2016. doi: 10.1007/978-3-319-24898-1.
    [6] H. Bethe, Zur theorie der metalle, Zeitschrift Für Physik, 71 (1931), 205-226.  doi: 10.1007/BF01341708.
    [7] L. Boßmann, Derivation of the 1d nonlinear Schrödinger equation from the 3d quantum many-body dynamics of strongly confined bosons, J. Math. Phys., 60 (2019), 031902, 30. doi: 10.1063/1.5075514.
    [8] L. BoßmannN. PavlovićP. Pickl and A. Soffer, Higher order corrections to the mean-field description of the dynamics of interacting Bosons, J. Stat. Phys., 178 (2020), 1362-1396.  doi: 10.1007/s10955-020-02500-8.
    [9] L. Boßmann and S. Teufel, Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons, Ann. Henri Poincaré, 20 (2019), 1003-1049.  doi: 10.1007/s00023-018-0738-7.
    [10] C. Brennecke and B. Schlein, Gross-Pitaevskii dynamics for Bose-Einstein condensates, Anal. PDE, 12 (2019), 1513-1596.  doi: 10.2140/apde.2019.12.1513.
    [11] R. Carles, Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9 (2011), 937-964.  doi: 10.4310/CMS.2011.v9.n4.a1.
    [12] T. ChenC. HainzlN. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, Comm. Pure Appl. Math., 68 (2015), 1845-1884.  doi: 10.1002/cpa.21552.
    [13] T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588.  doi: 10.1007/s00023-013-0248-6.
    [14] X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, Arch. Ration. Mech. Anal., 221 (2016), 631-676.  doi: 10.1007/s00205-016-0970-6.
    [15] X. Chen and J. Holmer, The derivation of the $\Bbb T^3$ energy-critical NLS from quantum many-body dynamics, Invent. Math., 217 (2019), 433-547.  doi: 10.1007/s00222-019-00868-3.
    [16] J. Chong, Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing NLS in $\mathbb{R}^3$, J. Math. Phys., 62 (2021), Paper No. 042106, 38 pp. doi: 10.1063/1.5099113.
    [17] S. DettmerD. HellwegP. RyyttyJ. J. ArltW. ErtmerK. SengstockD. S. PetrovG. V. ShlyapnikovH. KreutzmannL. Santos and M. Lewenstein, Observation of phase fluctuations in elongated bose-einstein condensates, Phys. Rev. Lett., 87 (2001), 160406. 
    [18] V. DunjkoV. Lorent and M. Olshanii, Bosons in cigar-shaped traps: Thomas-fermi regime, tonks-girardeau regime, and in between, Phys. Rev. Lett., 86 (2001), 5413-5416. 
    [19] L. ErdösB. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math., 59 (2006), 1659-1741.  doi: 10.1002/cpa.20123.
    [20] L. ErdösB. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614.  doi: 10.1007/s00222-006-0022-1.
    [21] L. ErdösB. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), 1099-1156.  doi: 10.1090/S0894-0347-09-00635-3.
    [22] L. ErdösB. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370.  doi: 10.4007/annals.2010.172.291.
    [23] L. Erdös and H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys., 5 (2001), 1169-1205.  doi: 10.4310/ATMP.2001.v5.n6.a6.
    [24] J. EsteveJ. B. TrebbiaT. SchummA. AspectC. I. Westbrook and I. Bouchoule, Observations of density fluctuations in an elongated bose gas: Ideal gas and quasicondensate regimes, Prl, 96 (2006), 130403. 
    [25] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, English edition, Classics in Mathematics, Springer, Berlin, 2007.
    [26] J. Fröhlich, T.-P. Tsai and H.-T. Yau, On a classical limit of quantum theory and the non-linear Hartree equation, GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal., Special Volume, Part Ⅰ, (2000), 57–78. doi: 10.1007/978-3-0346-0422-2_3.
    [27] M. Gaudin, The Bethe Wavefunction, Cambridge University Press, New York, 2014, Translated from the 1983 French original by Jean-Sébastien Caux. doi: 10.1017/CBO9781107053885.
    [28] J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. Ⅰ, Comm. Math. Phys., 66 (1979), 37-76.  doi: 10.1007/BF01197745.
    [29] J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. Ⅱ, Comm. Math. Phys., 68 (1979), 45-68.  doi: 10.1007/BF01562541.
    [30] M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Mathematical Phys., 1 (1960), 516-523.  doi: 10.1063/1.1703687.
    [31] F. Golse, On the dynamics of large particle systems in the mean field limit, In Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lect. Notes Appl. Math. Mech., Springer, [Cham], 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1.
    [32] M. Grillakis and M. Machedon, Pair excitations and the mean field approximation of interacting bosons, Ⅰ, Comm. Math. Phys., 324 (2013), 601-636.  doi: 10.1007/s00220-013-1818-7.
    [33] M. Grillakis and M. Machedon, Pair excitations and the mean field approximation of interacting bosons, Ⅱ, Comm. Partial Differential Equations, 42 (2017), 24-67.  doi: 10.1080/03605302.2016.1255228.
    [34] M. GrillakisM. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. Ⅱ, Adv. Math., 228 (2011), 1788-1815.  doi: 10.1016/j.aim.2011.06.028.
    [35] M. G. GrillakisM. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. Ⅰ, Comm. Math. Phys., 294 (2010), 273-301.  doi: 10.1007/s00220-009-0933-y.
    [36] B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv preprint, arXiv: 2003.05011.
    [37] K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277.  doi: 10.1007/BF01646348.
    [38] A. D. Jackson and G. M. Kavoulakis, Lieb mode in a quasi-one-dimensional bose-einstein condensate of atoms, Prl, 89 (2002), 070403. 
    [39] M. JeblickN. Leopold and P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation in two dimensions, Comm. Math. Phys., 372 (2019), 1-69.  doi: 10.1007/s00220-019-03599-x.
    [40] M. Jeblick and P. Pickl, Derivation of the time dependent two dimensional focusing NLS equation, J. Stat. Phys., 172 (2018), 1398-1426.  doi: 10.1007/s10955-018-2095-9.
    [41] R. KillipM. Vișan and X. Zhang, Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., 28 (2018), 1062-1090.  doi: 10.1007/s00039-018-0444-0.
    [42] K. KirkpatrickB. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130.  doi: 10.1353/ajm.2011.0004.
    [43] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185.  doi: 10.1007/s00220-008-0426-4.
    [44] A. Knowles and P. Pickl, Mean-field dynamics: Singular potentials and rate of convergence, Comm. Math. Phys., 298 (2010), 101-138.  doi: 10.1007/s00220-010-1010-2.
    [45] H. Koch and D. Tataru, Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., 167 (2018), 3207-3313.  doi: 10.1215/00127094-2018-0033.
    [46] M. LewinP. T. Nam and N. Rougerie, Derivation of nonlinear Gibbs measures from many-body quantum mechanics, J. Éc. Polytech. Math., 2 (2015), 65-115.  doi: 10.5802/jep.18.
    [47] M. LewinP. T. Nam and N. Rougerie, The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc., 368 (2016), 6131-6157.  doi: 10.1090/tran/6537.
    [48] E. H. Lieb, Exact analysis of an interacting Bose gas. Ⅱ. The excitation spectrum, Phys. Rev., 130 (1963), 1616-1624.  doi: 10.1103/PhysRev.130.1616.
    [49] E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. Ⅰ. The general solution and the ground state, Phys. Rev., 130 (1963), 1605-1616.  doi: 10.1103/PhysRev.130.1605.
    [50] E. H. Lieb and R. Seiringer, Proof of bose-einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88 (2002), 170409.  doi: 10.1103/PhysRevLett.88.170409.
    [51] E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, Oberwolfach Seminars, 34. Birkhäuser Verlag, Basel, 2005.
    [52] E. H. LiebR. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the gross-pitaevskii energy functional, Phys. Rev. A, 61 (2000), 043602.  doi: 10.1103/PhysRevA.61.043602.
    [53] E. H. LiebR. Seiringer and J. Yngvason, One-dimensional bosons in three-dimensional traps, PRL, 91 (2003), 150401. 
    [54] E. H. LiebR. Seiringer and J. Yngvason, One-dimensional behavior of dilute, trapped Bose gases, Comm. Math. Phys., 244 (2004), 347-393.  doi: 10.1007/s00220-003-0993-3.
    [55] E. H. Lieb and J. Yngvason, Ground state energy of the low density bose gas, Phys. Rev. Lett., 80 (1998), 2504-2507.  doi: 10.1103/PhysRevLett.80.2504.
    [56] D. Mendelson, A. R. Nahmod, N. Pavlović, M. Rosenzweig and G. Staffilani, Poisson commuting energies for a system of infinitely many bosons, arXiv preprint, arXiv: 1910.06959.
    [57] D. Mitrouskas, Derivation of Mean Field Equations and Their Next-Order Corrections: Bosons and Fermions, PhD thesis, LMU München, 2017.
    [58] P. T. Nam and M. Napiórkowski, Norm approximation for many-body quantum dynamics: Focusing case in low dimensions, Adv. Math., 350 (2019), 547-587.  doi: 10.1016/j.aim.2019.04.066.
    [59] M. Napiórkowski, Dynamics of interacting bosons: A compact review, arXiv preprint, arXiv: 2101.04594.
    [60] M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett., 81 (1998), 938-941. 
    [61] M. Olshanii and V. Dunjko, Short-distance correlation properties of the lieb-liniger system and momentum distributions of trapped one-dimensional atomic gases, PRL, 91 (2003), 090401. 
    [62] B. G. PachpatteInequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197. Academic Press, Inc., San Diego, CA, 1998. 
    [63] D. S. PetrovD. M. Gangardt and G. V. Shlyapnikov, Low-dimensional trapped gases, J. Phys. Ⅳ France, 116 (2004), 5-44.  doi: 10.1051/jp4:2004116001.
    [64] D. S. PetrovG. V. Shlyapnikov and J. T. M. Walraven, Regimes of quantum degeneracy in trapped 1D gases, Phys. Rev. Lett., 85 (2000), 3745-3749. 
    [65] P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation without positivity condition on the interaction, J. Stat. Phys., 140 (2010), 76-89.  doi: 10.1007/s10955-010-9981-0.
    [66] P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97 (2011), 151-164.  doi: 10.1007/s11005-011-0470-4.
    [67] P. Pickl, Derivation of the time dependent Gross-Pitaevskii equation with external fields, Rev. Math. Phys., 27 (2015), 1550003, 45 pp. doi: 10.1142/S0129055X15500038.
    [68] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
    [69] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, 2nd edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.
    [70] S. RichardF. GerbierJ. H. ThywissenM. HugbartP. Bouyer and A. Aspect, Momentum spectroscopy of 1D phase fluctuations in bose-einstein condensates, PRL, 91 (2003), 010405. 
    [71] I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Commun. Math. Phys., 291 (2009), 31-61.  doi: 10.1007/s00220-009-0867-4.
    [72] M. Rosenzweig, Mean-field convergence of point vortices without regularity, arXiv preprint, arXiv: 2004.04140.
    [73] M. Rosenzweig, Mean-field convergence of systems of particles with Coulomb interactions in higher dimensions without regularity, arXiv preprint, arXiv: 2010.10009.
    [74] N. Rougerie, Scaling limits of bosonic ground states, from many-body to nonlinear Schrödinger, EMS Surv. Math. Sci., 7 (2020), 253-408.  doi: 10.4171/emss/40.
    [75] B. Schlein, Derivation of effective evolution equations from microscopic quantum dynamics, In Evol. Equations, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 17 (2013), 511–572.
    [76] R. Seiringer and J. Yin, The Lieb-Liniger model as a limit of dilute bosons in three dimensions, Comm. Math. Phys., 284 (2008), 459-479.  doi: 10.1007/s00220-008-0521-6.
    [77] R. SeiringerJ. Yngvason and V. A. Zagrebnov, Disordered bose–einstein condensates with interaction in one dimension, Journal of Statistical Mechanics: Theory and Experiment, 2012 (2012), P11007.  doi: 10.1088/1742-5468/2012/11/p11007.
    [78] V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^3$ from the dynamics of many-body quantum systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1337-1365.  doi: 10.1016/j.anihpc.2014.09.005.
    [79] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.
    [80] T. Tao, Nonlinear Dispersive Equations, Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.
    [81] B. L. TolraK. M. O'HaraJ. H. HuckansW. D. PhillipsS. L. Rolston and J. V. Porto, Observation of reduced three-body recombination in a correlated 1D degenerate bose gas, Phys. Rev. Lett., 92 (2004), 190401. 
    [82] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz., 61 (1971), 118-134. 
  • 加载中

Tables(1)

SHARE

Article Metrics

HTML views(1696) PDF downloads(188) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return