Download e-book for iPad: Coherent Manipulation and Transport of Matter Waves by Wolfgang Arendt, Wolfgang P. Schleich

By Wolfgang Arendt, Wolfgang P. Schleich

ISBN-10: 3527408304

ISBN-13: 9783527408306

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It was justified (under certain conditions on Ω) by Ivrii [6] and Melrose [7] only in 1980. In 1915, Weyl concluded his work on the asymptotic behavior of eigenvalues with a study [8] of the elastic vibrations u of a homogeneous body with volume |Ω| which are determined by the solutions of the differential equation BΔu + A grad div u + λu = 0 . 8) Here λ is related to the frequency ν by λ = (2πν)2 , and A, B are positive constants (related to the Lamé coefficients characterizing the elastomechanical properties of the body).

Moreover, the signalling across synapses is a chemical phenomenon that is partially based on diffusion. Further Nobel Prizes in Medicine have been awarded for related discoveries in 1936, 1963, and 1970, in a crescendo that was made possible by the development of electron microscopy. A hundred years after Ramón y Cajal, diffusion processes still belong to the core of computational neuroscience – in a braid of chance, linear determinism, and chaos. Δ Δ Δ In 1887, Henri Poincaré won a contest sponsored by the King of Sweden asking for the solution of the famous three-body problem in celestial mechanics.

As an “illustration” he first considers “the case of a stretched √ string vibrating transversely” and derives the correct Weyl asymptotics N(λ) ~ λ, that is N(ν) ~ ν and thus D(ν) = constant (“when ν is large enough”). Then he continues: “When we pass from one dimension to three dimensions, and consider for example the vibrations of a cubic mass of air, we have (‘Theory of Sound’, paragraph 267) as the equation for ν2 , ν2 = p2 + q2 + r2 , where p, q, r are integers representing the number of subdivisions in the three directions.

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Coherent Manipulation and Transport of Matter Waves by Wolfgang Arendt, Wolfgang P. Schleich


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