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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Compare with :
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
If you are worried because
of convergence issues etc,
add to a small negative
: 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:
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.
See also: A. A. Terras & H.M. Stark , M. Kotani & T. Sunada
AAT : forthcoming book.
All periodic orbits, weighted by scattering amplitudes
Periodic orbits sum
(p) : “action” (function of )
ap : “stability amplitude”(function of )
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.
v (=40) regular graph :
Why? See previous lectures for an answer
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.