The spectral statistics of two closely related strongly chaotic quantum billiards are studied. Both are defined on the same triangular domain on the hyperbolic plane and differ only in the choice of the boundary conditions on the edges of the billiards. The fundamental domain is generated by the action of the reflection group ${\mathrm{T}}^{\mathrm{*}}$(2,3,8), which is an arithmetical group leading to an exponentially degenerate length spectrum of the classical periodic orbits. The boundary conditions on one billiard, called billard scrA, are chosen such that it does not belong to a representation of the reflection group, whereas for billiard scrB the boundary conditions correspond to an irreducible symmetry representation. The crucial property of arithmetical chaos, i.e., the exponential degeneracy of periodic orbits having the same length, is not affected by the choice of the boundary conditions. For both billiards our analysis of the spectral statistics is based on the first 1050 quantal energy levels, which we have computed using the boundary element method. It is found that the quantal level statistics for billiard scrB show the peculiar properties typical for arithmetical quantum chaos, as discovered previously for other arithmetical systems. Billiard scrA, however, behaves generically in that it shows at short- and mdeium-range correlations a behavior in agreement with the random-matrix theory. The periodic-orbit theory is scrutinized to shed some light on the mysterious differences between these two almost identical quantum billiards. The trace of the cosine-modulated heat kernel and the spectral form factor are studied. It is demonstrated that subtle properties of the characters attached to the classical periodic orbits are very important ingredients in the phenomenon of arithmetical quantum chaos.

• Received 21 November 1994

DOI:https://doi.org/10.1103/PhysRevE.51.4173