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Berard P.,Institute Fourier | Helffer B.,University Paris - Sud
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2014

Given a bounded open set in Rn (or in a Riemannian manifold), and a partition of ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk( ω) the infimum of maxj λ(ωj) over all k-partitions. A minimal k-partition is a partition that realizes the infimum. Although the analysis of minimal k-partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ(ωj) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition. © 2013 The Author(s) Published by the Royal Society. All rights reserved.

Jean-Francois A.,Institute Fourier
Nonlinearity | Year: 2012

We consider compact Lie group extensions of expanding maps of the circle, essentially restricting to SU(2) extensions. The main objective of the paper is the associated Ruelle transfer (or pull-back) operator F. Harmonic analysis yields a natural decomposition F = ⊕F j, where j indexes irreducible representation spaces. Using semi-classical techniques we extend a previous result by Faure proving an asymptotic spectral gap for the family {F j} when restricted to adapted spaces of distributions. Our main result is a fractal Weyl upper bound for the number of eigenvalues (the Ruelle resonances) of these operators out of some fixed disc centred on 0 in the complex plane. © 2012 IOP Publishing Ltd ∧ London Mathematical Society.

Mollard M.,Institute Fourier
Journal of Combinatorial Optimization | Year: 2014

Let γ(P m □P n ) be the domination number of the Cartesian product of directed paths P m and P n for m,n≥2. Liu et al. in (J. Comb. Optim. 22(4):651-662, 2011) determined the value of γ(P m □P n ) for arbitrary n and m≤6. In this work we give the exact value of γ(P m □P n ) for any m,n and exhibit dominating sets of minimum cardinality. © 2012 Springer Science+Business Media, LLC.

We consider a specific family of skew product of partially expanding map on the torus. We study the spectrum of the Ruelle transfer operator and show that in the limit of high frequencies in the neutral direction (this is a semiclassical limit), the spectrum develops a spectral gap, for a generic map. This result has already been obtained by Tsujii (2008 Ergodic Theory Dyn. Syst. 28 291-317). The novelty here is that we use semiclassical analysis which provides a different and quite natural description. We show that the transfer operator is a semiclassical operator with a well-defined 'classical dynamics' on the cotangent space. This classical dynamics has a 'trapped set' which is responsible for the Ruelle resonances spectrum. In particular, we show that the spectral gap is closely related to a specific dynamical property of this trapped set. © 2011 IOP Publishing Ltd & London Mathematical Society.

Faure F.,Institute Fourier | Sjostrand J.,University of Burgundy
Communications in Mathematical Physics | Year: 2011

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts. © 2011 Springer-Verlag.

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