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Search: MSC category 37K05 ( Hamiltonian structures, symmetries, variational principles, conservation laws )

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1. CMB 2011 (vol 55 pp. 858)

von Renesse, Max-K.
An Optimal Transport View of Schrödinger's Equation
We show that the Schrödinger 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:Schrödinger equation, optimal transport, Newton's law, symplectic submersion
Categories:81C25, 82C70, 37K05

2. CMB 2005 (vol 48 pp. 3)

Burq, N.
Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary according to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard and Leichtnam as well as Hassell and Zelditch, obtained under the additional assumption of the convexity of~$M$.

Categories:35Q55, 35BXX, 37K05, 37L50, 81Q20

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