Semiclassical non-concentration near hyperbolic orbits

By Hans Christianson. 2006

104 citations recorded by the source corpus.

For a large class of semiclassical pseudodifferential operators, including Schrodinger operators, P(h)=−h2Δg+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‖⩽C(log(1/h)/h)‖P(h)u‖+Clog(1/h)‖(I−A)u‖. This generalizes earlier estimates of Colin de Verdiere and Parisse [Y. Colin de Verdiere, B. Parisse, Equilibre instable en regime semi-classique: I – Concentration microlocale, Comm. Partial Differential Equations 19 (1994) 1535–1563; Equilibre instable en regime semi-classique: II – Conditions de Bohr–Sommerfeld, Ann. Inst. H. Poincare Phys. Theor. 61 (1994) 347–367] obtained for a special case, and of Burq and Zworski [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17 (2004) 443–471] for real hyperbolic orbits.

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Citations in the corpus

Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result (2008, unknown): “ A. SORIANO by Andrade et al. in [An1, An2] in the context of viscoelastic problems. For linear waves, we can mention the works due to Rauch and Taylor[Ra-Ta], Hitrik [HIT] and, recently Christianson [CHR]. For the nonlinear wave equation on compact manifolds with boundary, it is important to cite t”
Quantum decay rates in chaotic scattering (2007, background): “262 (2007), 145–195; ibid, Dispersive estimates for manifolds with one trapped orbit, preprint 2006, arXiv:math/0611845 [9] H. Christianson, Cutoff resolvent estimates and the semilinear Schrödinger equation, arXiv:0706.4315, to appear in Proc.”
Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result (2010, methodology): “For linear waves, we can mention the works due to Rauch and Taylor[Ra-Ta], Hitrik [HIT] and, recently Christianson [ CHR ].”
Sharp polynomial decay rates for the damped wave equation on the torus (2012, background): “1992], [Liu and Rao 2005], [Burq and Hitrik 2007; Nishiyama 2009] and [Christianson 2007; Schenck 2011] can be seen as counterparts for damped wave equations of the articles [Lebeau 1992], [Haraux 1989a; Jaffard 1990], [Burq and Zworski 2004] and [Anantharaman and Rivière 2012], respectively, in the”