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REDUCIBILITY FOR LINEAR QUASI PERIODIC DIFFERENTIAL EQUATIONS

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14 pages
REDUCIBILITY FOR LINEAR QUASI-PERIODIC DIFFERENTIAL EQUATIONS L. H. ELIASSON (Winter School, St Etienne de Tinee, February 7-11, 2011) We shall discuss the notion of reducibility (often referred to as Flo- quet theory) for linear quasi-periodic systems. Such systems occur both in linear theory (stationary Schrodinger equations with time-dependent potential) and in non-linear theory (variational equations for quasi- periodic solutions of non-linear systems) and reducibility is a natural notion – at least for quasi-periodic perturbations of constant coefficient systems. For ODE's we have a fairly complete picture of this property in the perturbative setting: under appropriate arithmetical and smoothness conditions, all systems are “almost reducible” and “almost all” systems are reducible. For PDE's our knowledge is much more sparse. We shall discuss some recent results proving reducibility for systems related to the Schrodinger equations with periodic boundary conditions. (1) Linear quasi-periodic systems (2) Reducibility (3) Reducibility and KAM (4) The local picture in finite dimension (5) Reducibility and the second Melnikov condition (6) Reducibility in infinite dimension – 3 examples (6.1) A linear Schrodinger equation on a torus Td with a quasi- periodic potential (6.2) A linear Schrodinger equation on R with a quasi-periodic potential (6.3) A variational equation for a quasi-periodic solution of a non-linear Schrodinger equation on a torus Td (7) About the proof of (6.1) Date: February 22

  • finite dimension

  • linear

  • d?i ?

  • td ?

  • almost all

  • system

  • periodic systems

  • there exist

  • schrodinger equation


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REDUCIBILITY FOR LINEAR QUASI-PERIODIC DIFFERENTIAL EQUATIONS
L. H. ELIASSON
(WinterSchool,StEtiennedeTine´e,February7-11,2011) We shall discuss the notion of reducibility (often referred to as Flo-quet theory) for linear quasi-periodic systems. Such systems occur both inlineartheory(stationarySchro¨dingerequationswithtime-dependent potential) and in non-linear theory (variational equations for quasi-periodic solutions of non-linear systems) and reducibility is a natural notion – at least for quasi-periodic perturbations of constant coefficient systems. For ODE’s we have a fairly complete picture of this property in the perturbative setting: under appropriate arithmetical and smoothness conditions, all systems are “almost reducible” and “almost all” systems are reducible. For PDE’s our knowledge is much more sparse. We shall discuss some recent results proving reducibility for systems related to theSchr¨odingerequationswithperiodicboundaryconditions. (1) Linear quasi-periodic systems (2) Reducibility (3) Reducibility and KAM (4) The local picture in finite dimension (5) Reducibility and the second Melnikov condition (6) Reducibility in infinite dimension – 3 examples (6.1)AlinearSchr¨odingerequationonatorus T d with a quasi-periodic potential (6.2)AlinearSchro¨dingerequationon R with a quasi-periodic potential (6.3) A variational equation for a quasi-periodic solution of a non-linearSchr¨odingerequationonatorus T d (7) About the proof of (6.1)
Date : February 22, 2011.
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