Abstract
AbstractWe prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Analysis
Reference25 articles.
1. Bérard, P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155, 249–276 (1977)
2. Besse, A.L.: Manifolds all of whose geodesics are closed, vol. 93 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Results in Mathematics and Related Areas]. With appendices by Epstein, D.B.A., Bourguignon, J.-P., Bérard-Bergery, L., Berger, M., Kazdan, J.L. Springer-Verlag, Berlin-New York (1978)
3. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit, vol. 268 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1999)
4. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)
5. Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. Éc. Norm. Supér. (4) 49, 543–577 (2016)