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Gibbs measure for the periodic derivative nonlinear Schr?dinger equation  [PDF]
Laurent Thomann,Nikolay Tzvetkov
Mathematics , 2010,
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.
Gibbs measure evolution in radial nonlinear wave and Schr?dinger equations on the ball  [PDF]
Jean Bourgain,Aynur Bulut
Mathematics , 2012,
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}.
Invariant Gibbs measures for the 2-d defocusing nonlinear Schr{?}dinger equations  [PDF]
Tadahiro Oh,Laurent Thomann
Mathematics , 2015,
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.
Invariant measure for the Schr?dinger equation on the real line  [PDF]
Federico Cacciafesta,Anne-Sophie de Suzzoni
Mathematics , 2014,
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 $L^2$.
Nonlocal Schr?dinger equations in metric measure spaces  [PDF]
Marcelo Actis,Hugo Aimar,Bruno Bongioanni,Ivana Gómez
Mathematics , 2015,
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.
Nonextensive thermodynamic functions in the Schr?dinger-Gibbs ensemble  [PDF]
J. L. Alonso,A. Castro,J. Clemente-Gallardo,J. C. Cuchí,P. Echenique-Robba,J. G. Esteve,F. Falceto
Physics , 2014,
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.
Asymptotics of the Weyl Function for Schr?dinger Operators with Measure-Valued Potentials  [PDF]
Annemarie Luger,Gerald Teschl,Tobias W?hrer
Mathematics , 2014, DOI: 10.1007/s00605-015-0740-9
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.
"Exact WKB integration'' of polynomial 1D Schr?dinger (or Sturm-Liouville) problem  [PDF]
A. Voros
Mathematics , 2002,
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.
Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schr?dinger equation  [PDF]
Giuseppe Genovese,Renato Lucà,Daniele Valeri
Mathematics , 2015,
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 $\int h_{2k}$, $k\geq1$.
Limit-Periodic Continuum Schr?dinger Operators with Zero Measure Cantor Spectrum  [PDF]
David Damanik,Jake Fillman,Milivoje Lukic
Mathematics , 2015,
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.
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