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Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E . Restrict it to some large cube Λ . Consider now I Λ , a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube Λ grows to infinity. We prove that, with probability one in the large volume...

Square functions, bounded analytic semigroups, and applications

Christian Le Merdy (2007)

Banach Center Publications

To any bounded analytic semigroup on Hilbert space or on L p -space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative L p -spaces, Banach lattices, and their subspaces. We give some applications to H ∞ functional calculus, similarity problems, multiplier theory, and control theory.

Stability for non-autonomous linear evolution equations with L p -maximal regularity

Hafida Laasri, Omar El-Mennaoui (2013)

Czechoslovak Mathematical Journal

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem ( P ) u ˙ ( t ) + A ( t ) u ( t ) = f ( t ) t -a.e. on [ 0 , τ ] , u ( 0 ) = 0 , where A : [ 0 , τ ] → ℒ ( X , D ) is a bounded and strongly measurable function and X , D are Banach spaces such that D → d ↪ X . Our main concern is to characterize L p -maximal regularity and to give an explicit approximation of the problem (P).

Stabilizability and controllability of systems associated to linear skew-product semiflows.

Mihail Megan, Adina Luminita Sasu, Bogdan Sasu (2002)

Revista Matemática Complutense

This paper is concerned with systems with control whose state evolution is described by linear skew-product semiflows. The connection between uniform exponential stability of a linear skew-product semiflow and the stabilizability of the associated system is presented. The relationship between the concepts of exact controllability and complete stabilizability of general control systems is studied. Some results due to Clark, Latushkin, Montgomery-Smith, Randolph, Megan, Zabczyk and Przyluski are generalized....

Stabilization of wave systems with input delay in the boundary control

Gen Qi Xu, Siu Pang Yung, Leong Kwan Li (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight ( 1 - μ ) is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert...

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