Let q(t) be a T-periodic potential such that ∫ T 0 q(t) dt < 0. The classical Lyapunov criterion to stability of Hill's equation -x + q(t)x = 0 is ∥q-∥ 1 = ∫ T 0 |q- (t)|dt < 4/T, where q- is the negative part of q. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using L α norms of q-, 1 < α < ∞. The numerical simulation to Mathieu's equation shows that the new criterion approximates the first stability region very well.

/pdf/a-lyapunov-type-stability-criterion-using-norms-t0g2y9a8rx.pdf

A Lyapunov-type stability criterion using ^{} norms

The spectral theory of periodic differential equations

/pdf/zony-neustoichivosti-odnomernykh-periodicheskikh-operatorov-cvw6n7gs9d.pdf

Зоны неустойчивости одномерных периодических операторов Шрeдингера и Дирака@@@Instability zones of periodic 1-dimensional Schrödinger and Dirac operators

Monograph project. I now revise the various chapters and will put new and longer versions on line from time to time. Comments and especially indications og references (unknown to me or forgotten) will be most welcome! Showkeys are left on since the numering is more likely to change than the labels from one version to another.

Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

We consider (not necessarily conservative) perturbations of a one phase Hamiltonian system written with action-angle variables where is real analytic, of the form where f, g are real analytic in all the variables and -periodic in . More generally we consider systems similar to system (*) with . It is well known that, using an iterated averaging process, it is posssible to eliminate `formally in ' the phase by a formal transformation tangent to the identity where are -periodic in . Then one obtains a formal autonomous system Fixing the normalization , we prove that the transform is Gevrey 1 in . As an application, we give a new proof of a result of Neishstadt: it is possible to represent the formal transformation by an actual transformation T (admitting as its asymptotic expansion) such that the transformation T reduces the system (*) to a system which is autonomous up to perturbations which are exponentially small in . It is possible to use a cut-off `at the smallest term' like Neishstadt but we prefer to use an incomplete Laplace transform. Then we obtain for T a nice dependence in and we improve Neishstadt's result. We will also give similar improvements for the basic adiabatic invariants theory. Our main statement generalizes a theorem of D Sauzin conjectured by P Lochak.

/pdf/gevrey-separation-of-fast-and-slow-variables-3ea5mg1830.pdf

Gevrey separation of fast and slow variables

On donne une majoration asymptotique, quand n tend vers l'infini, de la largeur du nieme intervalle d'instabilite pour l'equation de Hill a potentiel reel analytique. Pour les potentiels polynomes trigonometriques, on donne un developpement asymptotique sous la forme du module d'une somme finie de termes exponentiellement petits

/pdf/estimations-asymptotiques-des-intervalles-d-instabilite-pour-2atle68zxq.pdf

Estimations asymptotiques des intervalles d'instabilité pour l'équation de Hill

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Estimations asymptotiques des intervalles d'instabilité pour l'équation de Hill