By Wolfgang Arendt, Wolfgang P. Schleich
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Additional info for Coherent Manipulation and Transport of Matter Waves
It was justiﬁed (under certain conditions on Ω) by Ivrii  and Melrose  only in 1980. In 1915, Weyl concluded his work on the asymptotic behavior of eigenvalues with a study  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.
Coherent Manipulation and Transport of Matter Waves by Wolfgang Arendt, Wolfgang P. Schleich