No quantum ergodicity for star graphs

Details No quantum ergodicity for star graphs No quantum ergodicity for star graphs

2003-08-05

arxiv.org/abs/math-ph/0308005v1

Commun. Math. Phys. 250 259-285 (2004)

Physics

Mathematical Physics

26 pages, 5 figures

Scientific paper

10.1007/s00220-004-1145-0

We investigate statistical properties of the eigenfunctions of the Schrodinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These constructions are analogous to scars on short unstable periodic orbits.

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