The statistical description of irregular eigenfunctions: A semiclassical approach

We present a novel approach to study the statistical properties of eigenfunctions in quantum systems with chaotic classical counterpart. The method is based on a far reaching generalization of an old suggestion, made by Berry in 1977, saying that irregular eigenfunctions can be described as Gaussian-distributed random functions. The so-called Gaussian conjecture is supplemented with a well controlled approximation for the two-point spatial correlation function, the only microscopic input of the theory. The method employed to construct the correlation function makes use of the semiclassical expression for the quantum propagator in terms of classical trajectories due to Gutzwiller. After a short introduction, we present analytic and numerical evidence supporting the validity of the Gaussian conjecture and the power of the semiclassical two-point correlation function. We also discuss their experimental support and limitations. As a formal application of the resulting local Gaussian theory, we derive the results of two competing approaches, the so-called Random Wave and Ballistic Sigma models. Our conclusion is that, to date, the local Gaussian theory is the most general framework to describe the statistical properties of irregular eigenfunctions in clean (non-disordered) systems. Finally, we apply our ideas to the description of Coulomb Blockade conductance peaks in almost close quantum dots. We extend previous approaches and give a closed general expression for the average conductance. Our results are valid for any kind of confining potential and boundary conditions and are expressed in terms of universal coefficients and sums over classical trajectories.