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Counting nodal domains: The sequence of nodal counts

Counting nodal domains: The sequence of nodal counts. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A. This complexity is reflected in the nodal sequence!. An Appetizer: The geometrical contents of the nodal sequence. (The rectangle).

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Counting nodal domains: The sequence of nodal counts

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  1. Counting nodal domains: The sequence of nodal counts TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

  2. This complexity is reflected in the nodal sequence!

  3. An Appetizer: The geometrical contents of the nodal sequence. (The rectangle) (Idea due to D. Klawoon) Note: The nodal sequence is composed of integers and therefore only dimensionless quantities can be extracted. In other words, the geometry is determined up to isotropic scaling.

  4. Quantitative analysis – the distribution of normalized nodal counts G. Blum, S. Gnutzmann and U. Smilansky, Phys Rev Lett. 88, 114101 (2002)

  5. Separable systems • Example- the rectangle. Independent of a/b

  6. For separable problems in d > 2 : P() ¼ (1-  /m)(d-3)/2

  7. P() = [ 1- ( /2)2 ]-1/2 The Leading corrections (of order E-1/2 ) For the rectangular domain Can be computed byusing in the derivation

  8. Simple surfaces: n’’(m) 0 Simple surfaces of revolution Panos Karageorge and U.S. J. Phys. A: Math. Theor. 41 (2008) 205102 n(m) for a few ellipsoids n(m) m

  9. Bohr Sommerfeld (EBK) quantization Order the spectrum using the spectral counting function: The nodal count sequence :

  10. A trace formula for nodal counts (S. Gnutzmann, P. D. Karageorge and U Smilansky, Phys Rev Letters (2006) 97 090201.) Introduction:

  11. Useful formulae

  12. The leading (Weyl) term in the counting function:

  13. The next to leading term in the Weyl series (M=N=0).

  14. The (6,16) periodic manifold

  15. Unfolding the (6,16) trajectory Unfolding the (6,16) trajectory

  16. A trace formula for the nodal sequence – Separable systems Cumulative nodal counting For the rectangle, and using Poisson summation and the same methods as for the spectral trace formula: For detailed derivation see the file Rectangle.pdf

  17. Numerical demonstrations and investigations 1. Separable systems Ellipsoid of revolution The smooth terms: (c(k) – a k2)/k2 c(k)~a k2 k k k

  18. The oscillatory part= c(k) - smooth (k) Correct power-law for the envelope Scaled oscillatory part

  19. The scaled fluctuating part: Its Fourier transform =

  20. 2. Non separable (but classically integrable) systems Number of nodal intersections:

  21. Classically Chaotic Domains - Numerical investigation ( no theory for counting is yet available)

  22. The Sinai billiard

  23. Shortest unstable periodic orbits

  24. The “africa” domain: N=200 A few periodic orbits: A Preliminary length specvtrum From boundary intersections

  25. Can one count the shape of a drum? The nodal count sequence provides geometric information through a trace formula similar to the spectral trace formula. Is it equivalent to the geometrical contents of the spectral sequence?

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