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A trace formula for nodal counts: Surfaces of revolution

This research paper explores a trace formula for counting nodal counts on surfaces of revolution, focusing on separable systems and surfaces of revolution. The paper discusses the spectral counting function, nodal counting, and the geometric contents of the spectrum. The main feature is the checkerboard structure found in the nodal sequence. The research includes numerical simulations and examines the relationship between nodal counts and separable systems. The paper concludes with a discussion on nodal domains and their creation or merging through fission or fusion.

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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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