1. Tracing periodic orbits and cycles in the nodal count Amit Aronovitch, Ram Band, Yehonatan Elon, Idan Oren, Uzy Smilansky xz ( stands for the integral value of x) Fourier transforming the oscillating part of the nodal count of the following graph (Boundary conditions and lengths of bonds are indicated) gives the result: Introduction - billiards Introduction – metric graphs Classical – A particle which moves freely inside the billiard and is reflected from the walls Quantum – The eigenfunctions and the eigenvalues of the laplacian on the billiard: Classical – A particle which moves freely on the bonds and is probabilistically transmitted to any connected bond when it reaches a vertex Quantum - The eigenfunctions and the eigenvalues of the second derivative on the graph: Introduction – discrete graphs The connection between quantum and classical descriptions – trace formula The quantum spectral counting function is:. It can be presented as a sum of a smooth part and an oscillating part. [1,2] Quantum - The eigenvectors and the eigenvalues of the Laplacian matrix L: L i,j =L j,i <0 if vertices i and j are adjacent; L i,j =0 otherwise. The trace formula: ;. is the length of the classical periodic orbit. is a weight factor. Counting nodal domains … Nodal domains count – the number of connected domains where the n th wavefunction is of constant sign. For billiards we can also count the number of intersections of the nodal lines with the boundary (marked with ). … on billiards… on metric graphs … on discrete graphs The classical information stored in the nodal count – recent results -0.120.24-0.12 0.39 -0.78 0.39 -0.41 -0.41 -0.41 0.41 0.41 0.41 - + - -++ + + +-- - Fourier transforming to obtain - the power spectrum of Few classical periodic orbits of the Sinai billiard [1] M. G. Gutzwiller, in ”Chaos in Classical and Quantum Mechanics”, Vol. 1, Springer-Verlag, New York, 1990. [2] T.Kottos T and U. Smilansky 1999 Ann. Phys., NY 274 76 [3] G. Blum and U. Smilansky, Nodal Domains Statistics: A Criterion for Quantum Chaos, Phys Rev Letters, vol 88 (11), 2002. [4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I,Interscience Publishers, Inc., New York, N.Y., 1953. a a 2b 2c D N Courant’s Theorem: Nodal domains count of the n th eigenfunction is smaller or equal to n. [4] Define: and examine the distribution: The average taken over an ensemble of Laplacians. A priory,, where the three dots represents other parameters of the graph, such as: diameter, average degree, etc. For large enough graphs, we observed a data collapse: V = # of vertices B = # of bonds r = cycle dimension Example: The power spectrum contains lengths of periodic orbits as well as differences of lengths of bonds. Numerically, it was found that the formula for the nodal count sequence of this graph is: The nodal count and nodal intersection count of generic billiards are quantum mechanical properties. Current knowledge about them amounts to numerics and heuristic models for their smooth part. [3] Nevertheless, lengths of classical periodic orbits appear in the power spectrum of their oscillating part - as demonstrated below: lala lblb lclc 2la2la 2lb2lb 3la3la 2lc2lc 3lb3lb 4la4la 3lc3lc 4lb4lb 5la5la 3lb3lb 2lc2lc 4la4la 4lb4lb 5la5la lala lclc 2la2la Thus, the lengths of the classical periodic orbits can be deduced from the quantum spectral counting function.