The Uncertainty Principle.

If a function ip(x) is mostly concentrated in a box Q, while its Fourier transform $>(£) is concentrated mostly in Q', then we say ijj is microlocalized in Q X Q' in (x, £)-space. The uncertainty principle says that Q X Q' must have volume at least 1. We will explain what it means for ip to be microlocalized to more complicated regions S of volume ~ 1 in (x, £)space. To a differential operator P(x, D) is associated a covering of (x, £)-space by regions {Ba} of bounded volume, and a decomposition of L -functions u as a sum of "components" i t a microlocalized to BaThis decomposition u —» (uot) 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 Schrodinger operators. The theorems are joint work with D. H. Phong. CHAPTER I: THE SAK PRINCIPLE The uncertainty principle says that a function if), mostly concentrated in \x — Xo\ < 6X, cannot also have its Fourier transform «0 mostly concentrated in l£~~ £o| < % unless 8X -8^ > 1. This simple fact has far-reaching consequences for PDE, but until recently it was used only in a very crude form. The most significant classical application concerned the eigenvalues of a self adjoint differential operator A(x,D)= E «-(«(—Y \oc\<m \lOXJ with symbol A(x, £) = ]C|ai<m«()^According to the uncertainty principle, each box S = { ( x , O I | x x o | < « , | € & | < ^ 1 } should count for one eigenvalue, so the number of eigenvalues of A(x,D) which are less than K should be given approximately as the volume of the set S(A,K) = {(£,£) | A(x,£) < K}. If A is elliptic and K -+ oo, then this 'Volume-counting" is asymptotically correct (see Weyl [41], Carleman [5], Hörmander [23]). However, volume-counting can also produce grossly inaccurate estimates for systems as simple as two uncoupled harmonic oscillators. This is an expanded version of a Colloquium Lecture of the same title presented at the 89th annual meeting of the American Mathematical Society in Denver, Colorado, January 5-8, 1983; received by the editors February 1, 1983. 1980 Mathematics Subject Classification. Primary 35-02, 35H05, 35P15, 35S05, 42B20, 42B25, 81H05. 1 Supported by National Science Foundation Grant MCS 80-03072. © 1983 American Mathematical Society 0273-0979/83 $1.00 + $.25 per page 129