Mather measures associated with a class of Bloch wave functions. (English) Zbl 1256.81049
Ann. Henri Poincaré 13, No. 8, 1807-1839 (2012); erratum ibid. 15, No. 2, 415-417 (2014).
Summary: In this paper we study the Wigner transform for a class of smooth Bloch wave functions on the flat torus \(\mathbb T^n = \mathbb R^n /2\pi \mathbb Z^n\):
\[
\psi_{\hbar,P}(x) = a (\hbar,P,x) {\mathrm {e}}^{\frac{i}{\hbar} ( P\cdot x + \hat{v}(\hbar,P,x))}.
\]
On requiring that \(P \in \mathbb Z^n\) and \(\hbar = 1/N\) with \(N \in \mathbb N\), we select amplitudes and phase functions through a variational approach in the quantum states space based on a semiclassical version of the classical effective Hamiltonian \(\bar H(P)\) which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform of \(\psi_{\hbar,P}\) admits subsequences converging in the weak* sense to Mather probability measures on the phase space. These measures are invariant for the classical dynamics and Action minimizing.
MSC:
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
References:
| [1] | Ambrosio, L., Figalli, A., Friesecke, G., Giannoulis, J., Paul, T.: Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data. arXiv:1006.5388 · Zbl 1231.35186 |
| [2] | Ambrosio, L., Friesecke, G., Giannoulis, J.: Passage from quantum to classical molecular dynamics in the presence of Coulomb interactions. arXiv:0907.1205 · Zbl 1203.35192 |
| [3] | Anantharaman N.: (F-ENSLY) Entropy and the localization of eigenfunctions (English summary). Ann. Math. (2) 168(2), 435-475 (2008) · Zbl 1175.35036 · doi:10.4007/annals.2008.168.435 |
| [4] | Athanassoulis A., Paul T.: Smoothed affine Wigner transform. Appl. Comput. Harmon. Anal. 28(3), 313-319 (2010) · Zbl 1188.81124 · doi:10.1016/j.acha.2010.03.001 |
| [5] | Athanassoulis, A., Paul, T.: Strong phase-space semiclassical asymptotics. arXiv:1002.1371 · Zbl 1235.81099 |
| [6] | Aubin, J.-P., Ekeland, I.: Applied nonlinear analysis. Reprint of the 1984 original. Dover, Mineola (2006) · Zbl 0997.37042 |
| [7] | Aubry A.S., Le Daeron P.Y.: The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Phys. D 8(3), 381-422 (1983) · Zbl 1237.37059 · doi:10.1016/0167-2789(83)90233-6 |
| [8] | Barles, G.: Solutions de viscosité des quations de Hamilton-Jacobi. (French) [Viscosity solutions of Hamilton-Jacobi equations] Mathématiques et Applications (Berlin) [Mathematics and Applications], vol. 17. Springer, Berlin (1994) · Zbl 0819.35002 |
| [9] | Bernard P.: Existence of C1,1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds. Ann. Sci. École Norm. Sup. (4) 40(3), 445-452 (2007) · Zbl 1133.35027 |
| [10] | Bernard P.: On the number of Mather measures of Lagrangian systems. Arch. Ration. Mech. Anal. 197(3), 1011-1031 (2010) · Zbl 1247.70034 · doi:10.1007/s00205-009-0289-7 |
| [11] | Bernard P.: Symplectic aspects of Mather theory. Duke Math. J. 136(3), 401-420 (2007) · Zbl 1114.37036 |
| [12] | Bernard P., Buffoni B.: Optimal mass transportation and Mather theory. J. Eur. Math. Soc. 9(1), 85-121 (2007) · Zbl 1241.49025 · doi:10.4171/JEMS/74 |
| [13] | Billingsley P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics, New York (1999) · Zbl 0944.60003 · doi:10.1002/9780470316962 |
| [14] | Cagnetti F., Gomes D., Tran H.V.: Aubry-Mather measures in the non convex setting. SIAM J. Math. Anal. 43, 2601-2629 (2011) · Zbl 1258.35059 · doi:10.1137/100817656 |
| [15] | Dias Carneiro M.J.: On minimizing measures of the action of autonomous Lagrangians. Nonlinearity 8(6), 1077-1085 (1995) · Zbl 0845.58023 · doi:10.1088/0951-7715/8/6/011 |
| [16] | Chierchia, L.: KAM lectures. Dynamical systems. Part I, 155, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa (2003) · Zbl 1070.37044 |
| [17] | Contreras G., Iturriaga R., Paternain G.P., Paternain M.: Lagrangian graphs, minimizing measures and Mañe’s critical values. Geom. Funct. Anal. 8(5), 788-809 (1998) · Zbl 0920.58015 · doi:10.1007/s000390050074 |
| [18] | Combescure M., Ralston J., Robert D.: A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition. Commun. Math. Phys. 202(2), 463-480 (1999) · Zbl 0939.58031 · doi:10.1007/s002200050591 |
| [19] | Duistermaat J.J., Kolk J.A.C.: Distributions: Theory and Applications. Birkhäuser, Boston (2010) · Zbl 1213.46001 |
| [20] | Degli Esposti M., Graffi S., Isola S.: Classical limit of the quantized hyperbolic toral automorphisms. Commun. Math. Phys. 167(3), 471-507 (1995) · Zbl 0822.58022 · doi:10.1007/BF02101532 |
| [21] | Evans L.-C.: Further PDE methods for weak KAM theory. Calc. Var. Partial Differ. Equ. 35(4), 435-462 (2009) · Zbl 1179.35102 · doi:10.1007/s00526-008-0214-1 |
| [22] | Evans L.-C.: Towards a quantum analog of weak KAM theory. Commun. Math. Phys. 244(2), 311-334 (2004) · Zbl 1073.81037 · doi:10.1007/s00220-003-0975-5 |
| [23] | Evans, L.-C. (1-CA): Some new PDE methods for weak KAM theory (English summary). Calc. Var. Partial Differ. Equ. 17(2), 159-177 (2003) · Zbl 1032.37048 |
| [24] | Evans, L.-C.: Effective Hamiltonians and quantum states. Sminaire: Équations aux Dérivées Partielles, 2000-2001, Exp. No. XXII, 13 pp, École Polytech., Palaiseau (2001) · Zbl 1204.37063 |
| [25] | Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, vol. 88 · Zbl 0885.58022 |
| [26] | Fathi A., Siconolfi A.: Existence of C1 critical sub-solutions of the Hamilton-Jacobi equation. Invent. Math. 155(2), 363-388 (2004) · Zbl 1061.58008 · doi:10.1007/s00222-003-0323-6 |
| [27] | Fathi A., Giuliani A., Sorrentino A.: Uniqueness of invariant Lagrangian graphs in a homology or a cohomology class (English summary). Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(4), 659-680 (2009) · Zbl 1192.37086 |
| [28] | Gérard P., Markowich P., Mauser N.-J., Poupaud F.: Homogenization limits and Wigner transforms. Commun. Pure Appl. Math. 50(4), 323-379 (1997) · Zbl 0881.35099 · doi:10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C |
| [29] | Gérard, P.: Mesures semi-classiques et ondes de Bloch. Séminaire Équations aux dérivées partielles (1990-1991), Exposé No. 16 · Zbl 1061.58008 |
| [30] | Gérard, P.: A microlocal version of concentration-compactness. Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995), pp. 143-157. Progr. Nonlinear Differential Equations Appl., vol. 21. Birkhäuser Boston, Boston (1996) · Zbl 0868.35005 |
| [31] | Gérard P., Leichtnam E.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559-607 (1993) · Zbl 0788.35103 · doi:10.1215/S0012-7094-93-07122-0 |
| [32] | Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 Edition. Springer, Berlin (2001) · Zbl 1042.35002 |
| [33] | Gomes D.-A., Valls C.: Wigner measures and quantum Aubry-Mather theory. Asymptot. Anal. 51(1), 47-61 (2007) · Zbl 1216.81099 |
| [34] | Gomes D., Iturriaga R., Sánchez-Morgado H., Yu Y.: Mather measures selected by an approximation scheme. Proc. Am. Math. Soc. 138(10), 3591-3601 (2010) · Zbl 1204.37063 · doi:10.1090/S0002-9939-10-10361-X |
| [35] | Graffi, S., Paul, T.: Convergence of a quantum normal form and an exact quantization formula. arXiv:1102.0942 · Zbl 1248.81096 |
| [36] | Helffer B., Martinez A., Robert D.: Ergodicié et limite semi-classique. (French. English summary) [Ergodicity and the semiclassical limit]. Commun. Math. Phys. 109(2), 313-326 (1987) · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [37] | Lions P.-L., Paul T.: Sur les mesures de Wigner. (French) [On Wigner measures]. Rev. Mat. Iberoamericana 9(3), 553-618 (1993) · Zbl 0801.35117 · doi:10.4171/RMI/143 |
| [38] | Manè R.: On the minimizing measures of Lagrangian dynamical systems. Nonlinearity 5(3), 623-638 (1992) · Zbl 0799.58030 · doi:10.1088/0951-7715/5/3/001 |
| [39] | Manè R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9(2), 273-310 (1996) · Zbl 0886.58037 · doi:10.1088/0951-7715/9/2/002 |
| [40] | Markowich P., Paul T., Sparber C.: Bohmian measures and their classical limit. (English summary). J. Funct. Anal. 259(6), 1542-1576 (2010) · Zbl 1195.81060 · doi:10.1016/j.jfa.2010.05.013 |
| [41] | Mather J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21(4), 457-467 (1982) · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 |
| [42] | Mather J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207, 169-207 (1991) · Zbl 0696.58027 · doi:10.1007/BF02571383 |
| [43] | Mather J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier 43, 1349-1368 (1993) · Zbl 0803.58019 · doi:10.5802/aif.1377 |
| [44] | Paternain G.: Schrödinger operators with magnetic fields and minimal action functionals. Israel J. Math. 123, 1-27 (2001) · Zbl 0997.37042 · doi:10.1007/BF02784118 |
| [45] | Pulvirenti M.: Semiclassical expansion of Wigner functions. (English summary). J. Math. Phys. 47(5), 052-103 (2006) · Zbl 1111.81067 · doi:10.1063/1.2200143 |
| [46] | Ruzhansky M., Turunen V.: Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl. 16(6), 943-982 (2010) · Zbl 1252.58013 · doi:10.1007/s00041-009-9117-6 |
| [47] | Sorrentino, A.: Lecture notes on Mather’s theory for Lagrangian systems. ArXiv:1011.0590 · Zbl 1373.37144 |
| [48] | Sorrentino A., Viterbo C.: Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms. Geom. Topol. 14(4), 2383-2403 (2010) · Zbl 1207.37043 · doi:10.2140/gt.2010.14.2383 |
| [49] | Toth, J., Zelditch, S.: Norms of modes and quasi-modes revisited. (English summary). Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), pp. 435-458. Contemp. Math., vol. 320. Amer. Math. Soc., Providence (2003) · Zbl 1044.58038 |
| [50] | Zelditch, S.: Local and global analysis of eigenfunctions on Riemannian manifolds. (English summary). Handbook of Geometric Analysis, vol. 1, pp. 545-658. Adv. Lect. Math. (ALM), vol. 7. International Press, Somerville (2008) · Zbl 1176.58017 |
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