Results 1  10
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23
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 31 (7 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.
Nonselfadjoint differential operators
, 2002
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 30 (6 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.
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 12 (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.
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 5 (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.
Perturbations of selfadjoint operators with periodic classical flow
, 2003
"... We consider nonselfadjoint perturbations of a selfadjoint hpseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation satisfies h δ0 < ǫ ≤ ǫ0 for some δ0 ∈]0,1/2 [ and a su ..."
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Cited by 4 (2 self)
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We consider nonselfadjoint perturbations of a selfadjoint hpseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation satisfies h δ0 < ǫ ≤ ǫ0 for some δ0 ∈]0,1/2 [ and a sufficiently small ǫ0> 0. We get a complete asymptotic description of all eigenvalues in certain rectangles [−1/C,1/C] + iǫ[F0 − 1/C,F0 + 1/C]. In particular we are able to treat the case when ǫ> 0 is small but independent of h.
FROM RESOLVENT ESTIMATES TO DAMPED WAVES
"... Abstract. In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the undamped set. We show that if replacing the damping term with a higherorder complex absorbing potential gives an operato ..."
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Cited by 2 (0 self)
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Abstract. In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the undamped set. We show that if replacing the damping term with a higherorder complex absorbing potential gives an operator enjoying polynomial resolvent bounds on the real axis, then the “resolvent” associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding noncompact models. 1.
EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS
, 2012
"... We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of βdamped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset ..."
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Cited by 1 (0 self)
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We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of βdamped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of βdamped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.