A New Determinant for Quantum Chaos

Dynamical zeta functions [1], Fredholm determinants [2] and quantum Selberg zeta functions [3, 4] have recently been established as powerful tools for evaluation of classical and quantum averages in low dimensional chaotic dynamical systems [5] [8]. The convergence of cycle expansions [9] of zeta functions and Fredholm determinants depends on their analytic properties; particularly strong results exist for nice (Axiom A) hyperbolic systems, for which the dynamical zeta functions are holomorphic [10], and the Fredholm determinants are entire functions [11, 12]. In this note, motivated by the recent results of Eckhardt and Russberg [13], we conjecture that in contrast to the quantum Selberg zeta function, for nice hyperbolic systems the quantum Fredholm determinant (introduced below) is entire, i.e free of poles.