Peierls instability with electron-electron interaction: the commensurate case
V. Mastropietro
Communications on Pure & Applied Analysis 2002, 1(2): 135-159 doi: 10.3934/cpaa.2002.1.135
We consider a quantum many-body model describing a system of electrons interacting with themselves and hopping from one ion to another of a one dimensional lattice. We show that the ground state energy of such system, as a functional of the ionic configurations, has local minima in correspondence of configurations described by smooth $\frac{\pi}{pF}$ periodic functions, if the interaction is repulsive and large enough and pF is the Fermi momentum of the electrons. This means physically that a $d=1$ metal develop a periodic distortion of its reticular structure (Peierls instability). The minima are found solving the Eulero-Lagrange equations of the energy by a contraction method.
keywords: Quantum statistical physiscs renormalization group.
Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities
V. Mastropietro Michela Procesi
Communications on Pure & Applied Analysis 2006, 5(1): 1-28 doi: 10.3934/cpaa.2006.5.1
We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichelet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by a convergent perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.
keywords: tree formalism Diophantine and irrationality conditions Dirichlet boundary conditions. perturbation theory Lindstedt series method periodic solutions Nonlinear wave equation
Convergence of Lindstedt series for the non linear wave equation
G. Gentile V. Mastropietro
Communications on Pure & Applied Analysis 2004, 3(3): 509-514 doi: 10.3934/cpaa.2004.3.509
We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classical mechanics to study the existence of invariant tori in quasi-integrable Hamiltonian systems.
keywords: tree formalism Lindstedt series method Dirichlet boundary conditions Nonlinear wave equation perturbation theory Diophantine and irrationality conditions periodic solutions

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