×

zbMATH — the first resource for mathematics

Semiclassical non-concentration near hyperbolic orbits. (English) Zbl 1119.58018
J. Funct. Anal. 246, No. 2, 145-195 (2007); corrigendum ibid. 258, No. 3, 1060-1065 (2010).
Summary: For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, \(P(h)=-h^2\Delta_g+V(x)\), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if \(A\) is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then \[ \|u \|\leq C(\sqrt{\log(1/h)}/h\|P(h)u\}+C\sqrt{\log(1/h)}\|(I-A)u \|. \] This generalizes earlier estimates of Colin de Verdière and Parisse [Y. Colin de Verdière and B. Parisse [Commun. Partial Differ. Equations 19, No. 9–10, 1553–1563 (1994; Zbl 0819.35116) and Ann. Inst. Henri Poincaré, Phys. Théor. 61, No. 3, 347–367 (1994; Zbl 0845.35076)] obtained for a special case, and of Burq and Zworski [N. Burq and M. Zworski, J. Am. Math. Soc. 17, No. 2, 443–471 (2004; Zbl 1050.35058)] for real hyperbolic orbits.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
35P20 Asymptotic distributions of eigenvalues in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abraham, R.; Marsden, J., Foundations of mechanics, (1967), Benjamin New York
[2] Aurich, R.; Marklof, J., Trace formulae for 3-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard, Phys. D, 92, 1-2, 101-129, (1996) · Zbl 0890.58019
[3] Bony, J.-M.; Chemin, J.-Y., Espaces fonctionnels associés au calcul de weyl-Hörmander, Bull. soc. math. France, 122, 77-118, (1994) · Zbl 0798.35172
[4] Bony, J.-F.; Ramond, T.; Fujiie, S.; Zerzeri, M., Quantum monodromy for a homoclinic orbit
[5] Burq, N., Smoothing effect for Schrödinger boundary value problems, Duke math. J., 123, 2, 403-427, (2004) · Zbl 1061.35024
[6] Burq, N.; Zworski, M., Geometric control in the presence of a black box, J. amer. math. soc., 17, 443-471, (2004) · Zbl 1050.35058
[7] Colin de Verdière, Y.; Parisse, B., Équilibre instable en régime semi-classique: II - conditions de bohr – sommerfeld, Comm. partial differential equations, Ann. inst. H. Poincaré phys. theor., 61, 347-367, (1994) · Zbl 0845.35076
[8] Dimassi, M.; Sjöstrand, J., Spectral asymptotics in the semi-classical limit, (1999), Cambridge Univ. Press Cambridge · Zbl 0926.35002
[9] Duistermaat, J.J., Fourier integral operators, (1996), Birkhäuser Boston, MA · Zbl 0841.35137
[10] Evans, L.C.; Zworski, M., Lectures on semiclassical analysis
[11] Gérard, C.; Sjöstrand, J., Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. math. phys., 108, 391-421, (1987) · Zbl 0637.35027
[12] Guillemin, V., Wave – trace invariants, Duke math. J., 83, 2, 287-352, (1996) · Zbl 0858.58051
[13] Hitrik, M., Eigenfrequencies and expansions for damped wave equations, Methods appl. anal., 10, 4, 543-564, (2003) · Zbl 1088.58510
[14] Hofer, H.; Zehnder, E., Symplectic invariants and Hamiltonian dynamics, (1994), Birkhäuser Basel · Zbl 0837.58013
[15] Hörmander, L., Symplectic classification of quadratic forms, and general mehler formulas, Math. Z., 219, 413-449, (1995) · Zbl 0829.35150
[16] Iantchenko, A.; Sjöstrand, J., Birkhoff normal forms for Fourier integral operators II, Amer. J. math., 124, 817-850, (2002) · Zbl 1011.35144
[17] Iantchenko, A.; Sjöstrand, J.; Zworski, M., Birkhoff normal forms in semi-classical inverse problems, Math. res. lett., 9, 2-3, 337-362, (2002) · Zbl 1258.35208
[18] Klingenberg, W., Riemannian geometry, (1995), de Gruyter Berlin · Zbl 0911.53022
[19] Lebeau, G., Équation des ondes amorties, (), 73-109 · Zbl 0863.58068
[20] Lee, J.M., Introduction to smooth manifolds, (2003), Springer-Verlag New York
[21] Rauch, J.; Taylor, M., Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. pure appl. math., 28, 4, 501-523, (1975) · Zbl 0295.35048
[22] Sjöstrand, J., Semiclassical resonances generated by non-degenerate critical points, (), 402-429
[23] Sjöstrand, J., Geometric bounds on the density of resonances for semiclassical problems, Duke math. J., 60, 1, 1-57, (1990) · Zbl 0702.35188
[24] Sjöstrand, J., Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. res. inst. math. sci., 36, 573-611, (2000) · Zbl 0984.35121
[25] Sjöstrand, J., Resonances associated to a closed hyperbolic trajectory in dimension 2, Asymptot. anal., 36, 2, 93-113, (2003) · Zbl 1060.35096
[26] Sjöstrand, J.; Zworski, M., Quantum monodromy and semi-classical trace formulae, J. math. pures appl., 81, 1-33, (2002) · Zbl 1038.58033
[27] Sjöstrand, J.; Zworski, M., Fractal upper bounds on the density of semiclassical resonances · Zbl 1201.35189
[28] Tang, S.H.; Zworski, M., From quasimodes to resonances, Math. res. lett., 5, 261-272, (1998) · Zbl 0913.35101
[29] Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds, Adv. math., 6, 329-346, (1971) · Zbl 0213.48203
[30] Zelditch, S., Wave invariants for non-degenerate closed geodesics, Geom. funct. anal., 8, 1, 179-217, (1998) · Zbl 0908.58022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.