ON THE QUANTUM ZETA-FUNCTION

It is remarkable that the quantum zeta function, defined as a sum over energy eigenvalues E: Z(s) = Sigma 1/Es admits of exact evaluation in some situations for which not a single E be known. Herein we show how to evaluate instances of Z(s), and of an associated parity zeta funct ion Y(s), for various quantum systems. For some systems both Z(n), Y(n ) can be evaluated for infinitely many integers n. Such Z, Y values ca n be used, for example, to effect sharp numerical estimates of a syste m's ground energy. The difficult problem of evaluating the analytic co ntinuation Z(s) for arbitrary complex s is discussed within the contex ts of perturbation expansions, path integration, and quantum chaos.