Abstract:
In this paper we construct a Gibbs measure for the derivative Schr\"odinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.

Abstract:
We establish new results for the radial nonlinear wave and Schr\"odinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from \cite{B-T1,B-T2} and \cite {T1,T2}.

Abstract:
We consider the defocusing nonlinear Schr{\"o}dinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\mathbb{R}^2$. In particular, we discuss the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

Abstract:
In this paper, we build a Gibbs measure for the cubic defocusing Schr\"odinger equation on the real line with a decreasing interaction potential, in the sense that the non linearity $|u|^2u$ is multiplied by a function $\chi$ which we assume integrable and smooth enough. We prove that this equation is globally well-posed in the support of this measure and that the measure is invariant under the flow of the equation. What is more, the support of the measure (the set of initial data) is disjoint from $L^2$.

Abstract:
In this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic Schr\"odinger equations on spaces of homogeneous type. We prove the a.e. convergence when the initial data belongs to a dyadic version of an $L^2$ based Besov space. In particular we give a Haar wavelet characterization of these dyadic Besov spaces.

Abstract:
Schr\"odinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schr\"odinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distributions, and we prove that, when considered in composite quantum systems, it defines partition functions that do not factorize as products of partition functions of the noninteracting subsystems, even in the thermodynamical regime. This implies that it is not possible to define extensive thermodynamical magnitudes such as the free energy, the internal energy or the thermodynamic entropy by using these models. Therefore, we conclude that this distribution inspired by Schr\"odinger's idea can not be used to construct an appropriate quantum equilibrium thermodynamics.

Abstract:
We derive an asymptotic expansion for the Weyl function of a one-dimensional Schr\"odinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.

Abstract:
We review an "exact semiclassical" resolution method for the general stationary 1D Schr\"odinger equation with a polynomial potential. This method avoids having to compute any Stokes phenomena directly; instead, it basically relies on an elementary Wronskian identity, and on a fully exact form of Bohr--Sommerfeld quantization conditions which can also be viewed as a Bethe-Ansatz system of equations that will "solve" the general polynomial 1D Schr\"odinger problem.

Abstract:
We study the one dimensional periodic derivative nonlinear Schr\"odinger (DNLS) equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion $\int h_k$, $k\in \mathbb{Z}_{+}$. In each $\int h_{2k}$ the term with the highest regularity involves the Sobolev norm $\dot H^{k}(\mathbb{T})$ of the solution of the DNLS equation. We show that a functional measure on $L^2(\mathbb{T})$, absolutely continuous w.r.t. the Gaussian measure with covariance $(\mathbb{I}+(-\Delta)^{k})^{-1}$, is associated to each integral of motion $\int h_{2k}$, $k\geq1$.

Abstract:
We consider Schr\"odinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schr\"odinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $\lambda \in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant.