Linear and Nonlinear Theory of Eigenfunction Scars

Abstract 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 wave- functions. Quantitative measures of scarring are provided and comparisons are made with numerical data.

[1]  Wilkinson,et al.  Manifestations of classical chaos in the energy level spectrum of a quantum well. , 1995, Physical review letters.

[2]  Wintgen,et al.  Irregular wave functions of a hydrogen atom in a uniform magnetic field. , 1989, Physical review letters.

[3]  M. Berry,et al.  Closed orbits and the regular bound spectrum , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4]  F. Steiner,et al.  Quantum eigenstates of a strongly chaotic system and the scar phenomenon , 1995 .

[5]  Oriol Bohigas,et al.  Spectral properties of the Laplacian and random matrix theories , 1984 .

[6]  S. Fishman,et al.  Fredholm method for scars , 1996 .

[7]  O'Connor,et al.  Quantum localization for a strongly classically chaotic system. , 1988, Physical review letters.

[8]  O'Connor,et al.  Properties of random superpositions of plane waves. , 1987, Physical review letters.

[9]  Eric J. Heller,et al.  Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits , 1984 .

[10]  Eric J. Heller,et al.  Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier , 1992 .

[11]  G. Iooss,et al.  Chaotic behaviour of deterministic systems , 1983 .

[12]  S. Sridhar,et al.  Experimental observation of scarred eigenfunctions of chaotic microwave cavities. , 1991, Physical review letters.

[13]  N. Brenner,et al.  Semiclassical Wigner functions for quantum maps on a torus , 1995 .

[14]  E. Bogomolny.  Smoothed wave functions of chaotic quantum systems , 1988 .

[15]  A. Voros,et al.  Eigenstate structures around a hyperbolic point , 1996, chao-dyn/9609002.

[16]  Fishman,et al.  Semiclassical criterion for scars in wave functions of chaotic systems. , 1994, Physical review letters.

[17]  N. Balazs,et al.  The Quantized Baker's Transformation , 1987 .

[18]  K. Muller,et al.  Scars in wavefunctions of the diamagnetic Kepler problem , 1994 .

[19]  Stein,et al.  Experimental determination of billiard wave functions. , 1992, Physical review letters.

[20]  E. Heller.  Quantum intramolecular dynamics: Criteria for stochastic and nonstochastic flow , 1980 .

[21]  Lev Kaplan,et al.  Weak quantum ergodicity , 1998, chao-dyn/9810002.