( stands for the integral value of x)

1. 0.24 0.39 0.39 -0.12 -0.12 -0.78 - + + + + + - - - - + - N a 2b a D 2c la 2la lb 3la 5la 4la lc 2lb 4lb 2lc 3lb 3lc 4la 2lc lc la 2la 5la 4lb 3lb Introduction - billiards Introduction – metric graphs Introduction – discrete 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 eigenvectors and the eigenvalues of the Laplacian matrix L: Li,j=Lj,i<0 if vertices i and j are adjacent;Li,j=0 otherwise. • Quantum - The eigenfunctions and the eigenvalues of the second derivative on the graph: -0.41 0.41 0.41 -0.41 0.41 -0.41 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] • The trace formula: ; . is the length of the classicalperiodic orbit. is a weight factor. Few classical periodic orbits of the Sinai billiard Fourier transforming to obtain- the power spectrum of Thus, the lengths of the classical periodic orbits can be deduced from the quantum spectral counting function. Counting nodal domains … • Nodal domains count – the number of connected domains where the nth 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 discrete graphs … on billiards … on metric graphs The classical information stored in the nodal count – recent results Courant’s Theorem: Nodal domains count of the nth eigenfunction is smaller or equal to n. [4] Define: and examine the distribution: The average taken over an ensemble of Laplacians. Fourier transforming the oscillating part of the nodal count of the following graph(Boundary conditions and lengths of bonds are indicated) gives the result: 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] V = # of verticesB = # of bondsr = cycle dimension Nevertheless,lengths of classical periodic orbits appear in the power spectrum of their oscillating part -as demonstrated below: Example: 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: 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: ( stands for the integral value of x) [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.