The uncertainty principle

If a function ψ(x) is mostly concentrated in a box Q, while its Fourier transform (equation presented) is concentrated mostly in Q′, then we say ψ is microlocalized in Q × Q′ in (x, ξ) space. The uncertainty principle says that Q × Q′ must have volume at least 1. We will explain what it means for ψ to be microlocalized to more complicated regions B of volume ˜ 1 in (x, ξ)-space. To a differential operator P(x, D) is associated a covering of (x, ξ)-space by regions (B_α of bounded volume, and a decomposition of L²-functions u as a sum of “components” uα microlocalized to B_α. This decomposition u → (u_α) diagonalizes P(x, D) modulo small errors, and so can be used to study variable-coefficient differential operators, as the Fourier transform is used for constant-coefficient equations. We apply these ideas to existence and smoothness of solutions of PDE, construction of explicit fundamental solutions, and eigenvalues of Schrödinger operators. The theorems are joint work with D. H. Phong.