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61
Semiclassical Nonconcentration near Hyperbolic Orbits
"... Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆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 precis ..."
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Cited by 54 (19 self)
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Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆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 ‖ ≤ C ( √ log(1/h)/h)‖P(h)u ‖ + C √ log(1/h)‖(I − A)u ‖. This generalizes earlier estimates of Colin de VerdièreParisse [CVP] obtained for a special case, and of BurqZworski [BuZw] for real hyperbolic orbits. 1.
How to get a conservative wellposed linear system out of thin air
 Part I. Wellposedness and energy balance, ESAIM Control Optim. Calc. Var
"... Abstract. Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) (with the norm ‖z‖21/2 = 〈A0z, z〉) to another Hilbert space U. In Part I of this work we have proved that the system of equations z̈(t) +A0z(t) ..."
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Cited by 44 (7 self)
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Abstract. Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A1/20) (with the norm ‖z‖21/2 = 〈A0z, z〉) to another Hilbert space U. In Part I of this work we have proved that the system of equations z̈(t) +A0z(t) +
Dispersive Estimates for Manifolds with one Trapped Orbit
, 2006
"... For a large class of complete, noncompact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: ..."
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Cited by 32 (14 self)
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For a large class of complete, noncompact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator:
Applications of Cutoff Resolvent Estimates to the Wave Equation
, 2007
"... We consider solutions to the linear wave equation on noncompact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension. ..."
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Cited by 23 (6 self)
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We consider solutions to the linear wave equation on noncompact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension.
Optimal polynomial decay of functions and operator semigroups
 Math. Ann
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Spectral deviations for the damped wave equation
 G.A.F.A
, 2010
"... Abstract. We prove a Weyltype fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizo ..."
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Cited by 15 (0 self)
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Abstract. We prove a Weyltype fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the ‘entropy ’ that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyltype lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.
Energy decay for the damped wave equation under a pressure condition, preprint 2009
"... Abstract. We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we s ..."
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Cited by 15 (1 self)
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Abstract. We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis. hal00415529, version 1 10 Sep 2009 1.
Eigenfrequencies for damped wave equations on Zoll manifolds
, 2002
"... The eigenfrequencies associated to a damped wave equation are known to belong to a band parallel to the real axis. Under the assumption of periodicity of the geodesic flow we study the asymptotic distribution of the eigenfrequencies in the band. We show that the set of eigenfrequencies exhibits a c ..."
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Cited by 13 (6 self)
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The eigenfrequencies associated to a damped wave equation are known to belong to a band parallel to the real axis. Under the assumption of periodicity of the geodesic flow we study the asymptotic distribution of the eigenfrequencies in the band. We show that the set of eigenfrequencies exhibits a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. The asymptotics for the multiplicities of the clusters are also obtained.