Ergodic properties of eigenfunctions for the Dirichlet problem. (English) Zbl 0788.35103
The authors generalize to the Dirichlet-Laplacian a result already known for compact Riemannian manifolds and for semiclassical Schrödinger operators: namely that under ergodicity assumptions on the classical flow, there exists a subfamily of density 1 of orthonormalized eigenfunctions of the Laplacian, which are asymptotically equidistributed in phase space as the energy tends to infinity. The proof is based on refined versions of theorems on propagation of singularities for boundary value problems.
Reviewer: A.Martinez (Villetaneuse)
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
Keywords:
ergodicity; equidistribution of eigenfunction; Dirichlet-Laplacian; ergodicity assumptions; orthonormalized eigenfunctionsReferences:
| [1] | L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards , Comm. Math. Phys. 65 (1979), no. 3, 295-312. · Zbl 0421.58017 · doi:10.1007/BF01197884 |
| [2] | A.-M. Charbonnel, Comportement semi-classique des systèmes ergodiques , Ann. Inst. H. Poincaré Phys. Théor. 56 (1992), no. 2, 187-214. · Zbl 0752.35044 |
| [3] | Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien , Comm. Math. Phys. 102 (1985), no. 3, 497-502. · Zbl 0592.58050 · doi:10.1007/BF01209296 |
| [4] | A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators , Comm. Partial Differential Equations 3 (1978), no. 11, 979-1005. · Zbl 0389.35046 · doi:10.1080/03605307808820083 |
| [5] | D. Delande, Chaos in atomic and molecular physics , Chaos et Physique Quantique, Les Houches, 1989 eds. M. J. Giannori, A. Voros, and J. Zinn-Justin, North-Holland, Amsterdam, 1991, pp. 665-726. |
| [6] | J. Dodziuk, Eigenvalues of the Laplacian and the heat equation , Amer. Math. Monthly 88 (1981), no. 9, 686-695. JSTOR: · Zbl 0477.35072 · doi:10.2307/2320674 |
| [7] | P. Gérard, Microlocal defect measures , Comm. Partial Differential Equations 16 (1991), no. 11, 1761-1794. · Zbl 0770.35001 · doi:10.1080/03605309108820822 |
| [8] | P. Gérard, Mesures semi-classiques et ondes de Bloch , Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, École Polytech., Palaiseau, 1991, Exp. No. XVI, 19. · Zbl 0739.35096 |
| [9] | P. Grisvard, Elliptic Problems in Nonsmooth Domains , Monographs Stud. Math., vol. 24, Pitman, Boston, 1985. · Zbl 0695.35060 |
| [10] | B. Halpern, Strange billiard tables , Trans. Amer. Math. Soc. 232 (1977), 297-305. JSTOR: · Zbl 0374.53001 · doi:10.2307/1998942 |
| [11] | E. J. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits , Phys. Rev. Lett. 53 (1984), no. 16, 1515-1518. · doi:10.1103/PhysRevLett.53.1515 |
| [12] | J. W. Helton, An operator algebra approach to partial differential equations, propagation of singularities and spectral theory , Indiana Univ. Math. J. 26 (1977), no. 6, 997-1018. · Zbl 0373.35060 · doi:10.1512/iumj.1977.26.26081 |
| [13] | B. Helffer, A. Martinez, and D. Robert, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits , Phys. Rev. Lett. 53 (1984), no. 16, 1515-1518. · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [14] | 1 L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume 1 , Grundlehren Math. Wiss., vol. 256, Springer-Verlag, Berlin, 1983. · Zbl 0521.35001 |
| [15] | 2 L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume 3 , Grundlehren Math. Wiss., vol. 274, Springer-Verlag, Berlin, 1985. · Zbl 0601.35001 |
| [16] | K. Husimi, Some formal properties of the density matrix , Proc. Phys. Math. Soc. Japan 22 (1940), 264-314. · Zbl 0023.28601 |
| [17] | M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966), no. 4, part II, 1-23. · Zbl 0139.05603 · doi:10.2307/2313748 |
| [18] | A. Katok and J.-M. Strelcyn, Invariant Manifolds, Entropy, and Billiards; Smooth Maps with Singularities , Lecture Notes in Math., vol. 1222, Springer-Verlag, Berlin, 1986. · Zbl 0658.58001 |
| [19] | A. Knauf, Ergodic and topological properties of Coulombic periodic potentials , Comm. Math. Phys. 110 (1987), no. 1, 89-112. · Zbl 0616.58044 · doi:10.1007/BF01209018 |
| [20] | V. F. Lazutkin, On the asymptotics of the eigenfunctions of the Laplacian , Soviet Math. Dokl. 12 (1971), 1569-1572. · Zbl 0232.35075 |
| [21] | V. F. Lazutkin, The KAM Theory and Asymptotics of Spectrum of Elliptic Operators , Springer-Verlag, Berlin, 1991. · Zbl 0733.58038 |
| [22] | P. Leboef and A. Voros, Multiplicative formulation of quantum mechanics: a new setting for semiclassical analysis , in Analyse Algébrique des Perturbations Singulières, Proceedings, CIRM, Luminy, 1991, |
| [23] | S. W. McDonald and A. N. Kaufman, Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories , Phys. Rev. Lett. 42 (1979), 1189-1191. |
| [24] | R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems I , Comm. Pure Appl. Math. 31 (1978), no. 5, 593-617. · Zbl 0368.35020 · doi:10.1002/cpa.3160310504 |
| [25] | A. I. Shnirelman, Ergodic properties of eigenfunctions , Uspekhi Mat. Nauk 29 (1974), no. 6(180), 181-182. · Zbl 0324.58020 |
| [26] | A. I. Shnirelman, On the asymptotic properties of eigenfunctions in the regions of chaotic motion , addendum to [L2]. · Zbl 1296.35133 |
| [27] | L. Tartar, \(H\)-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations , Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 193-230. · Zbl 0774.35008 · doi:10.1017/S0308210500020606 |
| [28] | M. Taylor, Pseudodifferential Operators , Princeton Math. Ser., vol. 34, Princeton Univ. Press, Princeton, 1981. · Zbl 0453.47026 |
| [29] | A. Voros, Semi-classical approximations , Ann. Inst. H. Poincaré Sect. A (N.S.) 24 (1976), no. 1, 31-90. |
| [30] | E. Wigner, On the quantum correction for thermodynamic equilibrium , Phys. Rev. 40 (1932), 749. · Zbl 0004.38201 · doi:10.1103/PhysRev.40.749 |
| [31] | S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces , Duke Math. J. 55 (1987), no. 4, 919-941. · Zbl 0643.58029 · doi:10.1215/S0012-7094-87-05546-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
