Week 11
Week 11 overview 
 21. Trace formulas / 20 March 2018 /
 22. Spectral determinants / 22 March 2018 /
 Homework 11
 Optional
If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. The duality is made precise by means of trace formulas. The course is OVER. Trace formulae are beautiful, and there is nothing more to say. Just some mopping up to do.
Chapter 21  Trace formulas  
Dirac delta function  
Deterministic evolution  
Classical trace formula for maps  
Continuous time and Laplace transforms
Continuation of the compagnion video (week 9, lecture "Counting" on discrete time and generating functions, watch youtu.be/8DzhYfGNo1U). Discretetimegenerating functions compared with relating infinitesimal time evolution to the infinite time dynamics via a Laplace transform. The stage is set for the classical trace formula. (A paid video, recorded on a Wacom tablet and edited by Stephen Murphy, GaTech PE Interactive Instructional Media.) 

Contribution of a periodic orbit, continuous time  
Intrinsic coordinates  
Evaluation of the trace along the orbit[27 minutes]  
Trace formula  an interpretation  
Globallocal duality  
Don't compute the eigenfunctions 
We derive the spectral determinants, dynamical zeta functions. While traces and determinants are formally equivalent, determinants are the tool of choice when it comes to computing spectra. Skip sects. 20.5 and 20.6.
trace formula and spectral determinant Due 3 April 2018 

Discussion forum for week 11 