Maximum norms of chaotic quantum eigenstates and random waves

The growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated. The maximum norms provide a measure for localization effects in eigenfunctions. An upper bound for the maxima of random superpositions of random functions is derived. For the random-wave model this gives the bound c root lnE in the semiclassical limit E --> infinity. The growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic. The maximum norms of these systems are numerically shown to be conform with the random-wave model. Furthermore, the distribution of the locations of the maximum norms is discussed. (C) 1999 Elsevier Science B.V. All rights reserved.