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Understanding Nodal Counts: A Spectral Trace Formula for Surfaces of Revolution

This study presents a comprehensive exploration of the spectral trace formula relevant to counting nodal domains in quantum systems, specifically for surfaces of revolution. We investigate nodal counts through various historical perspectives, referencing key contributions from Sturm and Courant. The paper delves into the implications of the trace formula, examining both smooth and oscillatory components, as well as numerical simulations of the cumulative nodal counts. Special attention is given to the geometric aspects of the spectrum and the behavior of nodal sequences in separable systems.

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Understanding Nodal Counts: A Spectral Trace Formula for 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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