High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
Authors:
Dmitry Jakobson and Alexander Strohmaier
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 87-94
MSC (2000):
Primary 81Q50; Secondary 35P20, 37D30, 58J50, 81Q005
DOI:
https://doi.org/10.1090/S1079-6762-06-00164-8
Published electronically:
July 25, 2006
MathSciNet review:
2237272
Full-text PDF Free Access
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Abstract: 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.
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[BoK98]MR1644088 J. Bolte and S. Keppeler. Semiclassical time evolution and trace formula for relativistic spin-$1/2$ particles. Phys. Rev. Lett. 81 (1998), no. 10, 1987–1991.
[BoK99]MR1694732 J. Bolte and S. Keppeler. A semiclassical approach to the Dirac equation. Ann. Physics 274 (1999), no. 1, 125–162.
[Bol01]MR1838911 J. Bolte. Semiclassical expectation values for relativistic particles with spin 1/2. Invited papers dedicated to Martin C. Gutzwiller, Part III. Found. Phys. 31 (2001), no. 2, 423–444.
[BoG04]MR2073612 J. Bolte and R. Glaser. Zitterbewegung and semiclassical observables for the Dirac equation. J. Phys. A 37 (2004), no. 24, 6359–6373.
[BoG04.2]BoG04.2 J. Bolte, R. Glaser. A semiclassical Egorov theorem and quantum ergodicity for matrix-valued operators. Comm. Math. Phys. 247 (2004), no. 2, 391–419.
[Br75]MR0370660 M. Brin. Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9 (1975), 8–16.
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[Br82]MR0670078 M. Brin. Ergodic theory of frame flows. Ergodic Theory and Dynamical Systems II, Proc. Spec. Year, Maryland 1979-80, Progr. Math. 21, pp. 163–183, Birkhäuser, Boston, 1982.
[BrG80]MR0582702 M. Brin and M. Gromov. On the ergodicity of frame flows. Inv. Math. 60 (1980), 1–7.
[BrK84]MR0756723 M. Brin and H. Karcher. Frame flows on manifolds with pinched negative curvature. Comp. Math. 52 (1984), 275–297.
[BrP74]MR0343316 M. I. Brin and Ja. B. Pesin. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.
[BuP03]MR1988429 K. Burns and M. Pollicott. Stable ergodicity and frame flows. Geom. Dedicata 98 (2003), 189–210.
[CV85]MR0818831 Y. Colin de Verdière. Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102 (1985), 497–502.
[D82]MR661876 N. Dencker. On the propagation of polarization sets for systems of real principal type. J. Funct. Anal. 46 (1982), 351–372.
[EW96]EW96 G. Emmrich and A. Weinstein. Geometry of the transport equation in multicomponent WKB approximations. Comm. Math. Phys. 176 (1996), no. 3, 701–711.
[GMMP97]GMMP97 P. Gérard, P. Markowich, N. Mauser and F. Poupaud. Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997), no. 4, 323–379. Erratum: Comm. Pure Appl. Math. 53 (2000), no. 2, 280–281. ;
[San99]San99 M. R. Sandoval. Wave-trace asymptotics for operators of Dirac type. Comm. PDE 24 (1999), no. 9-10, 1903–1944.
[Shn74]MR0402834 A. I. Shnirelman. Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29 (1974), 181–182. (Russian)
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[Zel96]MR1384146 S. Zelditch. Quantum ergodicity of $C^*$ dynamical systems. Comm. Math. Phys. 177 (1996), 507–528.
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Additional Information
Dmitry Jakobson
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada
Email:
jakobson@math.mcgill.ca
Alexander Strohmaier
Affiliation:
Mathematisches Institut, Universität Bonn, Beringstrasse 1, D-53115 Bonn, Germany
MR Author ID:
654730
Email:
strohmai@math.uni-bonn.de
Keywords:
Dirac operator,
Hodge Laplacian,
eigenfunction,
frame flow,
quantum ergodicity
Received by editor(s):
April 26, 2006
Published electronically:
July 25, 2006
Additional Notes:
The first author was supported by NSERC, FQRNT and Dawson fellowship.
Communicated by:
S. Katok
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
