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Leading off-diagonal correction to the form factor of large graphs
Gregory Berkolaiko1, Holger Schanz, Robert S Whitney
1The Weizmann Institute of Science, Rehovot 76100, Israel.
Physical Review Letters
|March 23, 2002
Summary
We investigated quantum graphs using periodic-orbit theory. Pairs of self-intersecting orbits differing only in loop orientation contribute to the form factor, aligning with random-matrix theory for large graphs.
Area of Science:
- Quantum mechanics
- Mathematical physics
- Chaos theory
Background:
- Periodic-orbit theory is crucial for understanding quantum systems with chaotic classical dynamics.
- Investigating quantum graphs provides insights into the interplay between classical and quantum behavior.
Purpose of the Study:
- To analyze the form factor of a generic quantum graph with mixed classical dynamics and time-reversal symmetry.
- To extend periodic-orbit theory beyond the diagonal approximation.
Main Methods:
- Utilized periodic-orbit theory, going beyond the diagonal approximation.
- Calculated contributions from specific pairs of self-intersecting orbits.
Main Results:
- Identified pairs of self-intersecting orbits differing solely in loop orientation.
- These pairs yield a contribution of -2tau(2) to the form factor in the large graph limit.
Conclusions:
- The results confirm predictions from random-matrix theory for large quantum graphs.
- This work refines the application of periodic-orbit theory to complex dynamical systems.