Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics, and in physics. Despite this long tradition we are able to present a new rigorous result using only elementary calculus. In the case of the famous Bunimovich billiard table shown in Fig.1 we prove that the wave functions have… (More)
Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply… (More)
A well known result of Jaffard states that an arbitrary region on a torus controls , in the L 2 sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C ∞ .
Let f (z, ¯ z) be a positive bi-homogeneous hermitian form on C n , of degree m. A theorem proved by Quillen and rediscovered by Catlin and D'Angelo states that for N large enough, z, ¯ z N f (z, ¯ z) can be written as the sum of squares of homogeneous polynomials of degree m+N. We show this works for N ≥ C f ((n+m) log n) 3 where C f has a natural… (More)