  We prove that the largest eigenvalues of the beta ensembles of random matrix
theory converge in distribution to the low-lying eigenvalues of the random
Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the
positive half-line, where b_x' is white noise. In doing so we extend the
definition of the Tracy-Widom(beta) distributions to all beta>0, and also
analyze their tails. Last, in a parallel development, we provide a second
characterization of these laws in terms of a one-dimensional diffusion. The
proofs rely on the associated tridiagonal matrix models and a universality
result showing that the spectrum of such models converge to that of their
continuum operator limit. In particular, we show how Tracy-Widom laws arise
from a functional central limit theorem.
