A sharp lower bound for a resonance-counting function in even dimensions

Christiansen, T. J.

doi:10.5802/aif.3092

Cited by 8 publications

(6 citation statements)

References 33 publications

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“…Proof. Thanks to equation (2.3) in [Chr15] (which relies on the methods developed in [Zwo89]), we have that there exists C > 0 independent of h and n such that…”

Section: Proof Of Theoremmentioning

confidence: 99%

Equidistribution of phase shifts in trapped scattering

Ingremeau

2018

J. Spectr. Theory

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We consider semiclassical scattering for compactly supported perturbations of the Laplacian and show equidistribution of eigenvalues of the scattering matrix at (classically) non-degenerate energy levels. The only requirement is that sets of fixed points of certain natural scattering relations have measure zero. This extends the result of [GRHZ15], where the authors proved the equidistribution result under a similar assumption on fixed points but with the condition that there is no trapping. IntroductionConsider a Riemannian manifold (X, g) which is Euclidean near infinity, in the sense that there exist compact sets X 0 ⊂ X and K 0 ⊂ R d such that (X\X 0 , g) and (R d \K 0 , g eucl ) are isometric.Let us consider an operatorWe define the scattering matrix 1 S h : C ∞ (S d−1 ) −→ C ∞ (S d−1 ), which depends on h, by S h (φ in ) := e iπ(d−1)/2 φ out .The factor e iπ(d−1)/2 is taken so that the scattering matrix is the identity operator when (X, g) = (R d , g Eucl ) and V ≡ 0.For each h ∈ (0, 1], S h can be extended by density to a unitary operator acting on L 2 (S d−1 ). S h − Id is then a trace class operator. Therefore, S h admits a sequence of eigenvalues of modulus 1, which converge to 1, and which we denote by (e iβ h,n ) n∈N .Our aim in this paper will be to study the behaviour of (e iβ h,n ) in the limit where h → 0. To do this, we define a measure µ h on S 1 byfor any continuous f : S 1 −→ C. This measure is not finite, but µ h , f is finite as soon as 1 is not in the support of f .Let us now state the assumptions we make on the manifold X and on the potential V .

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“…Proof. Thanks to equation (2.3) in [Chr15] (which relies on the methods developed in [Zwo89]), we have that there exists C > 0 independent of h and n such that…”

Section: Proof Of Theoremmentioning

confidence: 99%

Equidistribution of phase shifts in trapped scattering

Ingremeau

2018

J. Spectr. Theory

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“…Let us mention here that the finiteness of R * , in this setting as well as in more general ones, has been applied to a variety of problems in scattering theory: see e.g. [St,Mi1,GuHaSi,Ch]. There are also well-known consequences for Schrödinger and wave evolution: see §1.2 below.…”

Section: Introductionmentioning

confidence: 98%

Exponential lower resolvent bounds far away from trapped sets

Datchev

Jin

2017

Preprint

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We give examples of semiclassical Schrödinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the resolvent kernel in detail using ordinary differential equation techniques. In particular, we identify a threshold spatial radius where the resolvent behavior changes. We apply these results to wave equations with radial wavespeed, identifying a corresponding threshold radius at which wave decay properties change.

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“…This is an active research area: see, for instance, [BT07,MMT08] and the papers cited therein. Furthermore, in the recent paper [Chr15], Christiansen used a resolvent bound of the form (1.4) to find a lower bound on the resonance counting function on even-dimensional Riemannian manifolds that are flat near infinity and contain a compactly supported perturbation.…”

Section: Introductionmentioning

confidence: 99%