Abstract
For real (time-reversal symmetric) quantum billiards, the mean length ⟨L⟩ of nodal line is calculated for the nth mode (n>>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. √n), 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 δL are of order √log k. For the curvature κ of nodal lines, ⟨|κ|⟩ and √⟨κ2⟩ are of order k, but ⟨|κ|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 k2 (i.e. n), the perimeter correction (again a long-range effect) is of order klog k (i.e. √nlog n) (and positive, notwithstanding nodal depletion near the boundary) and the fluctuations δN are of order k√log k. Generalizations of the results for mixed (Robin) boundary conditions are stated.