Nodal domains on quantum graphs

Details Nodal domains on quantum graphs Nodal domains on quantum graphs

2003-05-12

arxiv.org/abs/nlin/0305020v1

Nonlinear Sciences

Chaotic Dynamics

19 pages, uses IOP journal style files

Scientific paper

10.1088/0959-7174/14/1/011

We consider the real eigenfunctions of the Schr\"odinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds $B$. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.

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