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Search: All articles in the CMB digital archive with keyword SchrÃ¶dinger equation

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1. CMB Online first

Tang, Xianhua
 New super-quadratic conditions for asymptotically periodic SchrÃ¶dinger equation This paper is dedicated to studying the semilinear SchrÃ¶dinger equation $$\left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right.$$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results. Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov typeCategories:35J20, 35J60

2. CMB 2011 (vol 55 pp. 858)

von Renesse, Max-K.
 An Optimal Transport View of SchrÃ¶dinger's Equation We show that the SchrÃ¶dinger equation is a lift of Newton's third law of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = -\nabla^\mathcal W F(\mu)$ on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential $\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$ of the extended system and its Fisher information $\frac {\hbar^2} 8 \int |\nabla \ln \mu |^2 \,d\mu$. The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures. Keywords:SchrÃ¶dinger equation, optimal transport, Newton's law, symplectic submersionCategories:81C25, 82C70, 37K05
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