Sobolev Inequalities and Uncertainty Principles in Mathematical Physics. Part I

This is a first draft of lecture notes for a course given at the LMU Munich in July 2011. 1. Sobolev inequalities and uncertainty principles 1.1. Example: Stability of the hydrogen atom. We consider a single particle of mass 1/2 moving in three-dimensional space in the presence of a potential V . We recall that in classical mechanics the state of such a system (at a given time) is described by a point (p, x) ∈ R×R, where x denotes the particle position and p the particle momentum. The energy, which is conserved under the time evolution, is H(p, x) = p + V (x) . In quantum mechanics a state is described by a function ψ ∈ L2(R), a wave function, with ∫ R3 |ψ(x)| dx = 1 . (1.1) In view of this normalization requirement, the function |ψ| can be interpreted as the probability density of the position of the particle. On the other hand, by (1.1) and Plancherel’s theorem we know that the Fourier transform of ψ, ψ̂(p) = (2π)−3/2 ∫ R3 e−ip·xψ(x) dx , (1.2) satisfies a similar normalization condition ∫ R3 |ψ̂(p)| dp = 1 . (1.3) That is, |ψ̂| has the interpretation of the probability density of the momentum of the particle. The energy of the system in the state ψ is h[ψ] = ∫