Quantum ergodicity at a finite energy level

The purpose of this paper is to formulate the notion of quantum ergodicity at a finite energy level for certain quantum mechanics, by using the method of Sunada [Su1]. Under some assumptions on the corresponding classical mec hanics, we obtain a necessary and sufficient condition in terms of semi-cla ssical asymptotic behaviour of eigenfunctions of a quantum Hamiltonian so t hat the classical mechanics is ergodic. We also obtain a result on quantum weak mixing at a finite energy level which is a semiclassical analogue of t he notion introduced in [Z4].