Geodesic
High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
Jakobson, Dmitry1 ; Strohmaier, Alexander2
1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada
2 Mathematisches Institut, Universität Bonn, Beringstrasse 1, D-53115 Bonn, Germany
Electronic research announcements of the American Mathematical Society, Tome 12 (2006), pp. 87-94 Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators, and therefore the system remains noncommutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.
DOI : 10.1090/S1079-6762-06-00164-8

Jakobson, Dmitry 1 ; Strohmaier, Alexander 2

1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada
2 Mathematisches Institut, Universität Bonn, Beringstrasse 1, D-53115 Bonn, Germany 
@article{10_1090_S1079_6762_06_00164_8,
     author = {Jakobson, Dmitry and Strohmaier, Alexander},
     title = {High-energy limits of {Laplace-type} and {Dirac-type} eigenfunctions and frame flows},
     journal = {Electronic research announcements of the American Mathematical Society},
     pages = {87--94},
     year = {2006},
     volume = {12},
     doi = {10.1090/S1079-6762-06-00164-8},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-06-00164-8/}
}
TY  - JOUR
AU  - Jakobson, Dmitry
AU  - Strohmaier, Alexander
TI  - High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
JO  - Electronic research announcements of the American Mathematical Society
PY  - 2006
SP  - 87
EP  - 94
VL  - 12
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-06-00164-8/
DO  - 10.1090/S1079-6762-06-00164-8
ID  - 10_1090_S1079_6762_06_00164_8
ER  - 
%0 Journal Article
%A Jakobson, Dmitry
%A Strohmaier, Alexander
%T High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
%J Electronic research announcements of the American Mathematical Society
%D 2006
%P 87-94
%V 12
%U http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-06-00164-8/
%R 10.1090/S1079-6762-06-00164-8
%F 10_1090_S1079_6762_06_00164_8
Jakobson, Dmitry; Strohmaier, Alexander. High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows. Electronic research announcements of the American Mathematical Society, Tome 12 (2006), pp. 87-94. doi: 10.1090/S1079-6762-06-00164-8

[1] Arnol′D, V. I. Some remarks on flows of line elements and frames Dokl. Akad. Nauk SSSR 1961 255 257

[2] Bolte, Jens, Keppeler, Stefan Semiclassical time evolution and trace formula for relativistic spin-1/2 particles Phys. Rev. Lett. 1998 1987 1991

[3] Bolte, Jens, Keppeler, Stefan A semiclassical approach to the Dirac equation Ann. Physics 1999 125 162

[4] Bolte, Jens Semiclassical expectation values for relativistic particles with spin 1/2 Found. Phys. 2001 423 444

[5] Bolte, Jens, Glaser, Rainer Zitterbewegung and semiclassical observables for the Dirac equation J. Phys. A 2004 6359 6373

[6] Bolte, Jens, Glaser, Rainer A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators Comm. Math. Phys. 2004 391 419

[7] Brin, M. I. Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature Funkcional. Anal. i Priložen. 1975 9 19

[8] Brin, M. I. The topology of group extensions of 𝐶-systems Mat. Zametki 1975 453 465

[9] Brin, M. Ergodic theory of frame flows 1982 163 183

[10] Brin, M., Gromov, M. On the ergodicity of frame flows Invent. Math. 1980 1 7

[11] Brin, M., Karcher, H. Frame flows on manifolds with pinched negative curvature Compositio Math. 1984 275 297

[12] Brin, M. I., Pesin, Ja. B. Partially hyperbolic dynamical systems Izv. Akad. Nauk SSSR Ser. Mat. 1974 170 212

[13] Burns, Keith, Pollicott, Mark Stable ergodicity and frame flows Geom. Dedicata 2003 189 210

[14] Colin De Verdière, Y. Ergodicité et fonctions propres du laplacien Comm. Math. Phys. 1985 497 502

[15] Dencker, Nils On the propagation of polarization sets for systems of real principal type J. Functional Analysis 1982 351 372

[16] Emmrich, C., Weinstein, A. Geometry of the transport equation in multicomponent WKB approximations Comm. Math. Phys. 1996 701 711

[17] Gérard, Patrick, Markowich, Peter A., Mauser, Norbert J., Poupaud, Frédéric Homogenization limits and Wigner transforms Comm. Pure Appl. Math. 1997 323 379

[18] Sandoval, M. R. Wave-trace asymptotics for operators of Dirac type Comm. Partial Differential Equations 1999 1903 1944

[19] Šnirel′Man, A. I. Ergodic properties of eigenfunctions Uspehi Mat. Nauk 1974 181 182

[20] Lazutkin, Vladimir F. KAM theory and semiclassical approximations to eigenfunctions 1993

[21] Zelditch, Steven Uniform distribution of eigenfunctions on compact hyperbolic surfaces Duke Math. J. 1987 919 941

[22] Zelditch, Steven Quantum ergodicity of 𝐶* dynamical systems Comm. Math. Phys. 1996 507 528

Cité par Sources :