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# S + - PowerPoint PPT Presentation

5 . Scattering Approach (alternative secular equation). S -. S +. d=(j,i). t(d)=j. o(d)=i. Compare with :. 3. Spectral  function. Note : 1. 2 . The poles of det (I 2B - U(  )) coincide with the poles of detU.

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Presentation Transcript

d=(j,i)

t(d)=j

o(d)=i

Compare with :

3. Spectral  function

Note : 1.

2. The poles of det (I2B - U()) coincide with the poles of detU

3. The zeros of det (I2B - U()) coincide with the zeros of the

characteristic polynomial det ( IV - L)

4. The rhs of is a real and bounded function for real .

5. The last step is to expand det (I2B - U()) in periodic orbits

Comment:

If you are worried because

of convergence issues etc,

add to  a small negative

Imaginary part

: The set of primitive periodic orbits on G

ap() : amplitude

Combining and + periodic orbit product we get

1. All periodic orbits are included in the  function.

2. Functional equation for v regular graphs:

Define: and

3. Analogous to the  function for quantum graphs

4. Connects to a corresponding ( dependent) “classical” dynamics

in terms of the bi-stochastic matrix M

Md’,d() = | Ud’,d() |2 as for quantum graphs.

Compare with:

AAT : forthcoming book.

All periodic orbits, weighted by scattering amplitudes

3. Trace formula

Periodic orbits sum

(p) : “action” (function of )

ap : “stability amplitude”(function of )

“Weyl”

Explicit forms of the “actions”  (p) and the “stability amplitudes” can be written down

in terms and ap. They follow from the definition of the vertex scattering matrices

(i)d’,d given above.

Smooth+Fluctuations

(2-periodic orbits)

“Smooth”

v (=40) regular graph :

Why? See previous lectures for an answer

1-/v

A short list of open problems:

• Challenges for combinatorial graph theory.
• 1. Counting statistics of nb t-periodic cycles.
• 2. The Random Waves conjecture.
• - Other models of random graphs – The G(V,p) Ensemble.
• Localization, spectral measures.
• Isospectral graphs and the nodal count: Can nodal counts
• distinguish between isospectral graphs?

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