Quantum Classical Correspondence in Nonintegrable Systems with Continuous Parameter

We study quantum classical correspondence in nonintegrable systems with a continuous parameter. In a stadium billiard with an aspect ratio as a parameter, we show that the slope of the parametric motion of eigenvalues is mainly due to the first derivative of periodic orbit length with respect to the parameter. We also present a viewpoint for scarred wavefunctions in a continuous change of the systems. Since the discovery of scars 1 > in a stadium billiard/> a number of authors have studied properties of scars numerically in various systems.3> Several theoretical studies4 >,s> have tried to explain the extra accumulation of the wave function on an unstable periodic orbit (PO). Although these works seemed to confirm the existence of scars both numerically and theoretically, the semiclassical theory for individual eigenstates in nonintegrable systems has not been given yet. The difficulty in the investigation for the individual eigenstates is due to the strong interaction with other states. The origin of the repulsive interaction has been studied by many authors, and was related to chaotic behavior in the classical motion. 6> When we change a parameter in the Hamiltonian, it is well known that eigenvalues of nonintegrable systems show a number of avoided crossings due to the interaction. The definite representation for the interaction was given by the level dynamics,?> and the motion of levels was shown to be described by a classical Hamiltonian with complete integrability. The curvature of levels, i.e. the second derivative of eigenvalues with respect to the parameter, have been introduced to characterize the parametric property of eigenvalues, and the expressions for the large curvature tail of the distribution have been derived. 8> The universal behavior in large curvatures is checked in various systems by numerical calculations. 9 > On the other hand, the nonuniversal behavior of small curvatures was discovered numerically. 10> Zakrzewski and Delande11> suggest that the discrepancy in small curvatures can be used to classify the degree of scarring in different systems. In this paper, we consider the relation between classical PO's and the parametric motion of eigenvalues of the stadium billiard with an aspect ratio as a parameter. In § 2, we study properties of PO's when we change the parameter continuously. We concentrate on the continuity of the PO's and calculate the first derivative of the