Distribution of resonances for convex co-compact hyperbolic surfaces
Fractal Weyl laws for quantum resonances
Poisson formulæ for resonances.
Resonance expansions in wave propagation
Scattering matrix for asymptotically euclidean manifolds
Spectral asymptotics for manifolds with cylindrical ends
The spectrum of the Laplacian on manifolds with cylindrical ends consists of continuous spectrum of locally finite multiplicity and embedded eigenvalues. We prove a Weyl-type asymptotic formula for the sum of the number of embedded eigenvalues and the scattering phase. In particular, we obtain the optimal upper bound on the number of embedded eigenvalues less than or equal to , where is the dimension of the manifold.
Estimates on the number of scattering poles near the real axis for strictly convex obstacles
For the Dirichlet Laplacian in the exterior of a strictly convex obstacle, we show that the number of scattering poles of modulus in a small angle near the real axis, can be estimated by Const for sufficiently large depending on . Here is the dimension.
Correction and supplements to “Scattering matrices and scattering geodesics of locally symmetric spaces”
Scattering matrices and scattering geodesics of locally symmetric spaces
Elementary linear algebra for advanced spectral problems
We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical...
Control theory and high energy eigenfunctions
Variation de la phase de diffusion et distribution des résonances
Quantum decay rates in chaotic scattering
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Soliton scattering by delta impurities
Scattering matrix in conformal geometry
Control for Schrödinger operators on 2-tori: rough potentials
For the Schrödinger equation, on a torus, an arbitrary non-empty open set provides control and observability of the solution: . We show that the same result remains true for where , and is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for and conjectured for . The higher dimensional generalization remains open for .
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