LETTER TO THE EDITOR Nodal domain distributions for quantum maps

Abstract. The statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by Blum et al (Phys. Rev. Lett., 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., 88 (2002), 114102). These statistics were shown to be computable from the random wave model of the eigenfunctions. We here study the analogous problem for chaotic maps whose phase space is the two-torus. We show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory. We compare the predictions with the results of numerical computations involving quantum perturbed cat maps. PACS numbers: 05.45.Mt, 02.10.Yn Mathematics Subject Classification: 81Q50, 15A52 In a recent article Blum et al (2002) observed that the number-distributions of the nodal domains of quantum wavefunctions of billiards whose classical dynamics is integrable are different from those for chaotic billiards and argued that the latter are universal