Week 11
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Week 11 overview |
- 21. Trace formulas
- 22. Spectral determinants
- 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.
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Chapter 21 Trace formulas |
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Dirac delta function |
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Deterministic evolution |
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Classical trace formula for maps |
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Continuous time and Laplace transforms
Continuation of the compagnion video (week 9, lecture "Counting" on discrete time and generating functions, watch youtu.be/8DzhYfGNo1U). Discrete-timegenerating 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.) |
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Contribution of a periodic orbit, continuous time |
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Intrinsic coordinates |
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Evaluation of the trace along the orbit[27 minutes] |
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Trace formula - an interpretation |
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Global-local duality |
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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.
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trace formula and spectral determinant Due 12 April 2022 |
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Discussion forum for week 11 |