부산대학교 도서관

It was shown by the reviewer [Bull. Amer. Math. Soc. (N.S.) {\bf 25} (1991), no.~1, 55--60; MR1080004 (91j:58130)] that Selberg's zeta function for ${\rm PSL}(2,\bold Z)$ can be expressed in terms of the Fredholm determinants ${\rm det}(1\pm L_s)$ of the generalized transfer operator $L_s$ for the Gauss map $Tx=(1/x)\bmod 1$. This map serves to describe the geodesic flow on the modular surface $\scr H/{\rm PSL}(2,\bold Z)$ symbolically in the Bowen-Series approach. The thermodynamic formalism thus makes possible a completely new access to Selberg's theory without invoking the trace formula. In the physics literature the approach sketched above is the prime example for quantum chaos, which relates classical and quantum behaviour for chaotic systems. In the present paper the authors give a heuristic derivation of the above results. They show how the eigenfunctions of the hyperbolic Laplacian can be related to the eigenfunctions of the transfer operator $L_s$ for the eigenvalues $\pm 1$ via a certain Helgason transform. They also try to extend the approach to more general chaotic Hamiltonian systems (see also an earlier paper of Bogomolʹnyĭ\ [Nonlinearity {\bf 5} (1992), no.~4, 805--866; MR1174220 (93k:81057)]). It should be mentioned that J. Lewis has given a rigorous derivation of the relation between Maass waveforms and eigenfunctions of $L_s$ for eigenvalue $\lambda=1$. These functions turn out to be so-called period functions, already known in the theory of modular forms.