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Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

Michael Melgaard (2003)

Open Mathematics

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions...

Quasilinear waves and trapping: Kerr-de Sitter space

Peter Hintz, András Vasy (2014)

Journées Équations aux dérivées partielles

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...

Radiation conditions at the top of a rotational cusp in the theory of water-waves

Sergey A. Nazarov, Jari Taskinen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the linearized water-wave problem in a bounded domain (e.g.a finite pond of water) of ℝ 3 , having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point 𝒪 of the water surface, where a submerged body touches the surface (see Fig. 1). The radiation...

Radiation conditions at the top of a rotational cusp in the theory of water-waves

Sergey A. Nazarov, Jari Taskinen (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the linearized water-wave problem in a bounded domain (e.g. a finite pond of water) of ℝ 3 , having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point 𝒪 of the water surface, where a submerged body touches the surface (see Fig. 1)....

Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

Michael Hitrik, Johannes Sjöstrand (2008)

Annales scientifiques de l'École Normale Supérieure

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h -pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the...

Rational Krylov for nonlinear eigenproblems, an iterative projection method

Elias Jarlebring, Heinrich Voss (2005)

Applications of Mathematics

In recent papers Ruhe suggested a rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similarly to the Arnoldi method the search space is expanded by the direction from residual inverse iteration. Numerical methods demonstrate that the rational Krylov method can be accelerated considerably...

Recent results on Lieb-Thirring inequalities

Ari Laptev, Timo Weidl (2000)

Journées équations aux dérivées partielles

We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.

Recovering Asymptotics at Infinity of Perturbations of Stratified Media

Tanya Christiansen, Mark S. Joshi (2000)

Journées équations aux dérivées partielles

We consider perturbations of a stratified medium ℝ x n - 1 × ℝ y , where the operator studied is c 2 ( x , y ) Δ . The function c is a perturbation of c 0 ( y ) , which is constant for sufficiently large | y | and satisfies some other conditions. Under certain restrictions on the perturbation c , we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of c from knowledge of c 0 and the singularities of the scattering matrix at fixed energy....

Recovering the total singularity of a conormal potential from backscattering data

Mark S. Joshi (1998)

Annales de l'institut Fourier

The problem of recovering the singularities of a potential from backscattering data is studied. Let Ω be a smooth precompact domain in ℝ n which is convex (or normally accessible). Suppose V i = v + w i with v ∈ C c ∞ ( ℝ n ) and w i conormal to the boundary of Ω and supported inside Ω ‾ then if the backscattering data of V 1 and V 2 are equal up to smoothing, we show that w 1 - w 2 is smooth.

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