Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature

For real (time-reversal symmetric) quantum billiards, the mean length <L> of nodal line is calculated for the nth mode (n much greater than 1), with wavenumber k, using a Gaussian random wave model adapted locally to satisfy Dirichlet or Neumann boundary conditions. The leading term is of order k (i.e. rootn), and the first (perimeter) correction, dominated by an unanticipated long-range boundary effect, is of order log k (i.e. log n), with the same sign (negative) for both boundary conditions. The leading-order state-to-state fluctuations deltaL are of order rootlogk. For the curvature kappa of nodal lines, <\kappa\> and root<kappa(2)> are of order k, but <\kappa\(3)> and higher moments diverge. For complex (e.g. Aharonov-Bohm) billiards, the mean number <N> of nodal points (phase singularities) in the mode has a leading term of order k(2) (i.e. n), the perimeter correction (again a long-range effect) is of order k log k (i.e.,In log n) (and positive, notwithstanding nodal depletion near the boundary) and the fluctuations deltaN are of order krootlog k. Generalizations of the results for mixed (Robin) boundary conditions are stated.