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A note on eigenvalues of ordinary differential operators.

Alan Ho (1997)

Revista Matemática Iberoamericana

In this follow-up on the work of Fefferman-Seco [FS] an improved condition for the discrete eigenvalues of the operator -d2 / dx2 + V(x) is established for V(x) satisfying certain hypotheses. The eigenvalue condition in [FS] establishes eigenvalues of this operator to within a small error. Through an obervation due to C. Fefferman, the order of accuracy can be improved if a certain condition is true. This paper improves on the result obtained in [FS] by showing that this condition does indeed hold....

A WKB analysis of the Alfvén spectrum of the linearized magnetohydrodynamics equations

Manuel Núñez, Jesús Rojo (1993)

Applications of Mathematics

Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation div A 1 Δ u + A 2 u = 0 , where A 1 may have singularities in the domaind U of definition. We study the case when U is a half-plane and u possesses high Fourier components, analyzing the changes brought about by the singularity A 1 = . We show that absorptions of energy takes...

From multi-instantons to exact results

Jean Zinn-Justin (2003)

Annales de l’institut Fourier

In these notes, conjectures about the exact semi-classical expansion of eigenvalues of hamiltonians corresponding to potentials with degenerate minima, are recalled. They were initially motivated by semi-classical calculations of quantum partition functions using a path integral representation and have later been proven to a large extent, using the theory of resurgent functions. They take the form of generalized Bohr--Sommerfeld quantization formulae. We explain here their...

Mathematical models for laser-plasma interaction

Rémi Sentis (2005)

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

We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...

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