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

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

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

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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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