The theory of scarring of eigenfunctions of classically chaotic systems by short periodic orbits is extended in several ways. The influence of short-time linear recurrences on correlations and fluctuations at long times is emphasized. We include the contribution to scarring of nonlinear recurrences associated with homoclinic orbits and treat the different scenarios of random and nonrandom long-time recurrences. The importance of the local classical structure around the periodic orbit is emphasized, and it is shown for an optimal choice of test basis in phase space that scars must persist in the semiclassical limit. The crucial role of symmetry is also discussed which, together with the nonlinear recurrences gives a much improved account of the actual strength of scars for given classical orbits and in individual wavefunctions. Quantitative measures of scarring are provided and comparisons are made with numerical data.
1998 Academic Press I. INTRODUCTION Some years ago Berry [1] suggested that quantum eigenstates of classically chaotic systems should locally look like random superpositions of plane waves of the same (local) wavevector magnitude k, producing Gaussian random fluctuations in position space. This is the natural quantum manifestation of complete classical uniformity on the energy hypersurface for chaotic systems. It is also a consequence of random matrix theory (RMT), assuming that this theory applies to classically chaotic systems [2]. On the other hand, Gutzwiller periodic orbit theory of the energy spectra of classically chaotic systems has enjoyed much success, and it would be strange if periodic orbits had no visible manifestation in the eigenstates. Indeed, Article No. PH975773 171 0003-4916Γ98 25.00