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Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?

Laurent Miclo (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to [ 1 , 4 ] . The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity...

Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach.

Francis Nier (2004)

Journées Équations aux dérivées partielles

We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on 0 -forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of Δ f , h ( 0 ) and solves efficiently the question of weakly resonant wells.

Quantum diffusion and generalized Rényi dimensions of spectral measures

Jean-Marie Barbaroux, François Germinet, Serguei Tcheremchantsev (2000)

Journées équations aux dérivées partielles

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order p at time T for the state ψ defined by [ 1 T ∫ 0 T ∥ | X | p / 2 e - i t H ψ ∥ 2 d t ] . We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure μ ψ associated to the hamiltonian H and the state ψ . We especially concentrate on continuous models.

Quantum graph spectra of a graphyne structure

Ngoc T. Do, Peter Kuchment (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

We study the dispersion relations and spectra of invariant Schrödinger operators on a graphyne structure (lithographite). In particular, description of different parts of the spectrum, band-gap structure, and Dirac points are provided.

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...

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