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Absolute continuity of the spectrum of periodic operators of mathematical physics

Tatiana Suslina (2000)

Journées équations aux dérivées partielles

The lecture is devoted to the problem of absolute continuity of the spectrum of periodic operators. A general approach to this problem was suggested by L. Thomas in 1973 for the case of the Schrödinger operator with periodic electric potential. Further application of his method to concrete operators of mathematical physics met analytic difficulties. In recent years several new problems in this area have been solved. We propose a survey of known results in this area, including very recent, and formulate...

Absolutely continuous spectrum and scattering in the surface Maryland model

François Bentosela, Philippe Briet, Leonid Pastur (2001)

Journées équations aux dérivées partielles

We study the discrete Schrödinger operator H in 𝐙 d with the surface quasi periodic potential V ( x ) = g δ ( x 1 ) tan π ( α · x 2 + ω ) , where x = ( x 1 , x 2 ) , x 1 ∈ 𝐙 d 1 , x 2 ∈ 𝐙 d 2 , α ∈ 𝐑 d 2 , ω ∈ [ 0 , 1 ) . We first discuss a proof of the pure absolute continuity of the spectrum of H on the interval [ - d , d ] (the spectrum of the discrete laplacian) in the case where the components of α are rationally independent. Then we show that in this case the generalized eigenfunctions have the form of the “volume” waves, i.e. of the sum of the incident plane wave and reflected from the hyper-plane 𝐙 d 1 waves, the form...

Around the bounded L 2 curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel (2008)

Journées Équations aux dérivées partielles

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation □ g φ = 0 , where ≫ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L 2 bounds on the curvature tensor R of ≫ is a major step towards the proof of the bounded L 2 curvature conjecture.

Asymptotic behavior of scattering amplitudes in semi-classical and low energy limits

Didier Robert, H. Tamura (1989)

Annales de l'institut Fourier

We study the semi-classical asymptotic behavior as ( h → 0 ) of scattering amplitudes for Schrödinger operators - ( 1 / 2 ) h 2 Δ + V . The asymptotic formula is obtained for energies fixed in a non-trapping energy range and also is applied to study the low energy behavior of scattering amplitudes for a certain class of slowly decreasing repulsive potentials without spherical symmetry.

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