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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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slide2

d=(j,i)

t(d)=j

o(d)=i

Compare with :

slide3

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

slide4

Combining and + periodic orbit product we get

Comments:

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.

slide5

Compare with:

See also: A. A. Terras & H.M. Stark , M. Kotani & T. Sunada

AAT : forthcoming book.

All periodic orbits, weighted by scattering amplitudes

slide6

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

slide7

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?
slide8

היום קצר, והמלאכה מרובה ....

לא עליך המלאכה לגמור, ולא אתה בן חורין להבטל ממנה.

The day is short and the work is aplenty…

it is not up to you to finish it, but you are not free to remain idle.