APPLIED MATH SEMINAR

Title: "Ergodicity of quantum eigenfunctions in classically chaotic systems"

Speaker: Alex Barnette, Courant Instructor in Applied Mathematics, Courant Institute

Abstract: it is a long-standing question how quantum (wave) eigenfunctions behave in the semiclassical (short-wavelength) limit, when the corresponding classical dynamics is chaotic. For instance, do they become ergodic (equidistributed across space) or are remnants ('scars') of periodic orbits important? Recent analytic results in manifolds of uniform negative curvature have led to a conjecture by Rudnick and Sarnak that every eigenfunction becomes ergodic. We study a point particle inside a 2D cavity ('billiard'), whose quantum equivalent is the familiar Laplacian eigenproblem, namely the cavity's resonant modes. We numerically investigate the rate of equidistribution and compare to semiclassical estimates involving classical correlation functions. We have collected thousands of modes at energy ranges well beyond those in the literature; I will outline the numerical innovations which make this possible. These methods are orders of magnitude faster than standard boundary integral methods, and should find applications to resonance problems in acoustics and electromegnetism.