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Hart F. Smith, Maciej Zworski (0)

Annales de l’institut Fourier

Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David Borthwick, Colin Guillarmou (2016)

Journal of the European Mathematical Society

On geometrically finite hyperbolic manifolds Γ ∖ ℍ d , including those with non-maximal rank cusps, we give upper bounds on the number N ( R ) of resonances of the Laplacian in disks of size R as R → ∞ . In particular, if the parabolic subgroups of Γ satisfy a certain Diophantine condition, the bound is N ( R ) = 𝒪 ( R d ( log R ) d + 1 ) .

Wave Operators for Defocusing Matrix Zakharov-Shabat Systems with Pnonvanishing at Infinity

Demontis, Francesco, der Mee, Cornelis van (2010)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 34L25; secondary: 47A40, 81Q10.In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ± ∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.

Weak Asymptotics for Schrödinger Evolution

S. A. Denisov (2010)

Mathematical Modelling of Natural Phenomena

In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decaying potential.

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

Plamen Stefanov (2001)

Journées équations aux dérivées partielles

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...

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