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 (Bα of bounded volume, and a decomposition of L2-functions u as a sum of “components” uα microlocalized to Bα. This decomposition u → (uα) 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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics