Curved boundary corrections to nodal line statistics in chaotic billiards

A Gaussian random wavefunction that satisfies Dirichlet and Neumann conditions locally on a convex circular boundary is introduced. The average of the square of the wavefunction and its derivatives are computed and their asymptotics studied in the semi-classical limit. The mean nodal line length (L) is calculated and the first order boundary effect shown to be of order log k, where k is the wavenumber. In the limit of vanishing boundary curvature (large boundary radius) these results are shown to approach those for a straight wall boundary.