From quasimodes to resonances

Tang, Siu-Hung; Zworski, Maciej

doi:10.4310/mrl.1998.v5.n3.a1

Cited by 98 publications

(128 citation statements)

References 12 publications

(24 reference statements)

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“…If the resonance z 0 satisfies |Im z 0 | = O(h M ) with M >> 1, one can apply the semiclassical maximum principle and the a priori exponential estimate of the modified resolvent (P θ −z) −1 obtained by Tang-Zworski [22]. Using these techniques, Stefanov [21] has proved that Proposition 6.1 (Stefanov) Assume that V is compactly supported and let E 0 > 0.…”

Section: Estimate On the Spectral Projectormentioning

confidence: 99%

“…The proof is based on the a priori estimate of the resolvent given by Tang-Zworski [22] and on the fact that the number of resonances in…”

Section: Estimate On the Spectral Projectormentioning

confidence: 99%

“…In [20] and [15], the method employed stands on the semiclassical maximum principle of Tang and Zworski [22] and a resolvent estimate of Burq [2].…”

Section: Introductionmentioning

confidence: 99%

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Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

Bony

Michel

2004

Communications in Mathematical Physics

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We obtain some microlocal estimates of the resonant states associated to a resonance z 0 of an h-differential operator. More precisely, we show that the normalized resonant states are O( |Im z 0 |/h +h ∞ ) outside the set of trapped trajectories and are O(h ∞ ) in the incoming area of the phase space.As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such bound in some examples.

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Section: Estimate On the Spectral Projectormentioning

confidence: 99%

“…The proof is based on the a priori estimate of the resolvent given by Tang-Zworski [22] and on the fact that the number of resonances in…”

Section: Estimate On the Spectral Projectormentioning

confidence: 99%

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Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

Bony

Michel

2004

Communications in Mathematical Physics

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“…The rest of this section is devoted to the proof of Theorem 3.1. We follow the approach of Tang and Zworski [41] and we use the constructions of [2, Section 4] (see also Christianson [10] for hyperbolic orbits), where the propagation of singularities through a hyperbolic fixed point is studied, and of [1, Section 3], where a sharp estimate for the weighted resolvent for real energies is given.…”

Section: Resolvent Estimatementioning

confidence: 99%

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Bony¹,

Fujiié²,

Ramond³

et al. 2011

Ann. inst. Fourier

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“…, z p } with the same multiplicities. We need the following estimate due to Zworski [Z2] (in this generality, see the proof of Lemma 1 in [TZ1])…”

Section: Propositionmentioning

confidence: 99%

Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Stefanov¹

2001

Journées Équations Aux Dérivées Partielles

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We study semiclassical resonances in a box Ω(h) of height h N , N 1. We show that the semiclassical wave front set of the resonant states (including the "generalized eigenfunctions") is contained in the set T of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # (h) with discrete spectrum the number of resonances in Ω(h) is bounded by the number of eigenvalues of P # (h) in an interval a bit larger than the projection of Ω(h) on the real line. As an application, we prove a Weyl type estimate of the number of resonances in Ω(h) in terms of the measure of T . We prove a similar estimate in case of classical scattering by a metric and obstacle.

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From quasimodes to resonances

Cited by 98 publications

References 12 publications

Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

Microlocalization of Resonant States and Estimates of the Residue of the Scattering Amplitude

Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Weyl type upper bounds on the number of resonances near the real axis for trapped systems

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