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Spectral Density Tracing with Periodic Orbits in Canonical Trace Formula for Counting Walks and Spectra

The trace formula expresses the spectral density in terms of periodic orbits. Explore how the spectrum of A can count periodic walks and vice versa. Discover the relationship between periodic walks and the spectrum of A. Learn about t-periodic non-backscattering walks, Kesten McKay, and Non-Ramanujan graphs. Dive into the w-trace formula and backscattering walks weighted by (1-w) per backscatter. Delve into the spectrum of equilateral quantum graphs from Bartholdi.

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Spectral Density Tracing with Periodic Orbits in Canonical Trace Formula for Counting Walks and Spectra

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  1. 2. The trace formula Purpose: express the spectral density in terms of “periodic orbits”

  2. 2.a The “canonical “ trace formula The spectrum of A can be used to count n.b. periodic walks and vice versa The counts of n.b. periodic walks can be used to compute the spectrum of A

  3. The number of t-periodic non backscattering walks :

  4. Sum over t-periodic non-backscattering walks. Kesten McKay

  5. Non Ramanujan graphs :

  6. 2.b A continuous family of trace formulas Y(w) gives a weight (1-w) to back scattering!

  7. The w - trace formula includes counting of backscattering walks weighted by (1-w) per backscatter!

  8. 2.c The spectrum of equilateral quantum graphs from Bartholdi

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