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On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set Ω ⊂ ℝ n , with Dirichlet boundary conditions. The observation is done on a subset ω of Lebesgue measure | ω | = L | Ω | , where L ∈ ( 0 , 1 ) is fixed. We denote by 𝒰 L the class of all possible such subsets. Let T > 0 . We consider first the benchmark problem of maximizing...

On the Bethe-Sommerfeld conjecture

Leonid Parnovski, Alexander V. Sobolev (2000)

Journées équations aux dérivées partielles

We consider the operator in ℝ d , d ≥ 2 , of the form H = ( - Δ ) l + V , l > 0 with a function V periodic with respect to a lattice in ℝ d . We prove that the number of gaps in the spectrum of H is finite if 8 l > d + 3 . Previously the finiteness of the number of gaps was known for 4 l > d + 1 . Various approaches to this problem are discussed.

On the convergence of SCF algorithms for the Hartree-Fock equations

Eric Cancès, Claude Le Bris (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

On the curvature and torsion effects in one dimensional waveguides

Guy Bouchitté, M. Luísa Mascarenhas, Luís Trabucho (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the Laplace operator in a thin tube of ℝ 3 with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube's cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube's central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies.

On the distribution of resonances for some asymptotically hyperbolic manifolds

R. G. Froese, Peter D. Hislop (2000)

Journées équations aux dérivées partielles

We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute S -matrix that is unitary for real values of the energy. This paramatrix is the S -matrix for a model laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance...

On the distribution of scattering poles for perturbations of the Laplacian

Georgi Vodev (1992)

Annales de l'institut Fourier

We consider selfadjoint positively definite operators of the form - Δ + P (not necessarily elliptic) in ℝ n , n ≥ 3 , odd, where P is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if { λ j } ( Im λ j ≥ 0 ) are the scattering poles associated to the operator - Δ + P repeated according to multiplicity, it is proved that for any ϵ > 0 there exists a constant...

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