Mathematical study of scattering resonances

Zworski, Maciej

doi:10.1007/s13373-017-0099-4

Cited by 101 publications

(120 citation statements)

References 279 publications

(512 reference statements)

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“…The latter class includes scattering by several convex obstacles (see Figure 7), where spectral gaps have been observed in microwave scattering experiments by Barkhofen et al [B * 13]. We refer to the reviews of Nonnenmacher [No11] and Zworski [Zw17] for an overview of results on spectral gaps for open quantum chaotic systems.…”

Section: Applications Of Fupmentioning

confidence: 98%

An introduction to fractal uncertainty principle

Dyatlov

2019

Journal of Mathematical Physics

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Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex co-compact hyperbolic surfaces.A fractal uncertainty principle (FUP) is a statement in harmonic analysis which can be vaguely formulated as follows (see Figure 1):No function can be localized in both position and frequency close to a fractal set.

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Section: Applications Of Fupmentioning

confidence: 98%

An introduction to fractal uncertainty principle

Dyatlov

2019

Journal of Mathematical Physics

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“…When iterating (67) forwardly, we cannot control the asymptotic behaviour and, in general, the growing mode will dominate. Our aim is to introduce a new set of linear independent sequences {I ( ) k } ∞ k=−m still satisfying the recurrence relation (67) but in a way that we remove the growing asymptotic behaviour 10 .…”

Section: A Taylor Series Expansionsmentioning

confidence: 99%

Hyperboloidal slicing approach to quasinormal mode expansions: The Reissner-Nordström case

2018

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We study quasi-normal modes of black holes, with a focus on resonant (or quasi-normal mode) expansions, in a geometric frame based on the use of conformal compactifications together with hyperboloidal foliations of spacetime. Specifically, this work extends the previous study of Schwarzschild in this geometric approach to spherically symmetric asymptotically flat black hole spacetimes, in particular Reissner-Nordström. The discussion involves, first, the non-trivial technical developments needed to address the choice of appropriate hyperboloidal slices in the extended setting as well as the generalization of the algorithm determining the coefficients in the expansion of the solution in terms of the quasi-normal modes. In a second stage, we discuss how the adopted framework provides a geometric insight into the origin of regularization factors needed in Leaver's Cauchy based foliations, as well as into the discussion of quasi-normal modes in the extremal black hole limit.

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“…Let ∆ = N j=1 ∂ 2 x j be the Laplacian operator in the complex Lebesgue space L 2 C (R m ) with odd m ≥ 1. For operators H obtained as various types of perturbations of (−∆) on compact subsets of R m , resonances k are defined as poles of the resolvent (H − z 2 ) −1 extended in a generalized sense through R into the lower complex halfplane C − := {z ∈ C : Im z < 0} (this and other types of definitions can be found, e.g., in [4,17,55,56]).…”

Section: Main Goals and Related Studiesmentioning

confidence: 99%

“…The collection of all resonances Σ(H) ⊂ C that are associated with an operator H (in short, resonances of H) is a multiset, i.e., a set in which an element e can be repeated a finite number m e ∈ N of times (this number m e is called the multiplicity of e). The multiplicity of a resonance k is defined as the multiplicity of the corresponding generalized pole of (H − z 2 ) −1 (e.g., [17,56]) or as the multiplicity of a certain analytic function built from the resolvent of H and generating resonances as its zeros ([4, 13, 14]).…”

Section: Main Goals and Related Studiesmentioning

confidence: 99%

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On the multilevel internal structure of the asymptotic distribution of resonances

Albeverio

Karabash

2019

Journal of Differential Equations

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We prove that the asymptotic distribution of resonances has a multilevel internal structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions in R 3 , quantum graphs, and 1-D photonic crystals. In the case of N ≥ 2 point interactions, the set of resonances Σ(H) essentially consists of a finite number of sequences with logarithmic asymptotics. We show how the leading parameters µ of these sequences are connected with the geometry of the set Y = {y j } N j=1 of interaction centers y j ∈ R 3 . The minimal parameter µ min corresponds to the sequences with 'more narrow' and so more observable resonances. The asymptotic density of such narrow resonances can be expressed via the multiplicity of µ min , which occurs to be connected with the symmetries of Y and naturally introduces a finite number of classes of configurations of Y . In the case of quantum graphs and 1-D photonic crystals, the decomposition of Σ(H) into a finite number of asymptotic sequences is proved under additional commensurability conditions. To address the case of a general quantum graph, we introduce families of special asymptotic density functions for two classes of strips in C. The obtained results and effects are compared with those of obstacle scattering.

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Mathematical study of scattering resonances

Abstract: We provide an introduction to mathematical theory of scattering resonances and survey some recent results.

Cited by 101 publications

References 279 publications

An introduction to fractal uncertainty principle

An introduction to fractal uncertainty principle

Hyperboloidal slicing approach to quasinormal mode expansions: The Reissner-Nordström case

On the multilevel internal structure of the asymptotic distribution of resonances

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