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A trace formula for nodal counts: Surfaces of revolution

A trace formula for nodal counts: Surfaces of revolution. Sven Gnutzmann Panos Karageorgi U. S. Rehovot, April 2006. Reminder: The spectral trace formula or how to count the spectrum. The spectral counting function: . Trace formula :.  Smooth.  Oscillatory. A periodic orbit.

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A trace formula for nodal counts: Surfaces of revolution

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  1. A trace formula for nodal counts: Surfaces of revolution Sven Gnutzmann Panos Karageorgi U. S. Rehovot, April 2006

  2. Reminder: The spectral trace formula or how to count the spectrum

  3. The spectral counting function: Trace formula :  Smooth  Oscillatory A periodic orbit The geometrical contents of the spectrum

  4. The sequenceofnodal counts n =20 n=8 Sturm (1836) : For d=1 : n = n Courant (1923) : For d>1 : n n

  5. Counting Nodal Domains: Separable systems Rectangle, Disc “billiards” in R2 Surfaces of revolution Liouville surfaces Main Feature – Checkerboard structure

  6. Simple Surfaces of Revolution (SSR)  for a few ellipsoids n(m) simple surfaces: n’’(m) 0 m

  7. Bohr Sommerfeld (EBK) quantization

  8. Nodal counting Order the spectrum using the spectral counting function: The nodal count sequence : The cumulative nodal count:

  9. Cmod(k) C(k)

  10. A trace formula for the nodal sequence Cumulative nodal counting

  11. Numerical simulation: the smooth term Ellipsoid of revolution (c(k) – a k2)/k2 c(k)~a k2 k k k

  12. The fluctuating part = c(k) - smooth (k) Correct power-law

  13. The scaled fluctuating part: Its Fourier transform = the spectrum of periodic orbits lengths

  14. The main steps in the derivation Poisson summation Semi-classical (EBK) n+1/2 ! n

  15. Change of variables: Approximate: Integration limit: Another change of variables

  16. The oscillatory term Saddle point integration: Picks up periodic tori with action: Collecting the terms one gets the trace formula

  17. Closing remarks : What is the secret behind nodal counts for separable systems? Consider the rectangular billiard: E(n,m)= n2 +  m2 ;  (n,m)= n m ~ (Lx / Ly)2 Follow the nodal sequence as a function of  : At every rational value of  there will be pairs of integers (n1,m1) and (n2,m2) for which the eigen-values cross:  -+ E (n1,m1) < E (n2,m2) ; E (n1,m1) = E (n2,m2) ; E (n1,m1) > E (n2,m2) ! at this  the nodal sequence will be swapped ! Thus: The swaps in the nodal sequence reflect the the value of ! Geometry of the boundary

  18. Gnutzmann films presents Nodal domains are created or merged by fission or fusion

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