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Geometric Characterization of Nodal Patterns and Domains

This research aims to characterize billiards by studying the geometrical features of their nodal domains. Investigating the Helmholtz equation on 2D surfaces with Dirichlet Boundary conditions allows for analysis of the total number of nodal domains and their distribution functions. The study delves into the impact of energy intervals and dimensionless parameters on the distribution functions. By exploring the geometry of wave functions and their relationship with classical trajectories, the research identifies universal features across different surfaces, including rectangles, discs, and surfaces of revolution. Analyzing separable surfaces and random waves further sheds light on the distinct features and behaviors of nodal domains in various systems. The study also explores open questions regarding the connection between classical trajectories and wave functions, analytic and statistical derivations of the genus distribution, and the implications for chaotic billiards.

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Geometric Characterization of Nodal Patterns and Domains

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  1. Geometric Characterization of Nodal Patterns and Domains Y. Elon, S. Gnutzman, C. Joas U. Smilansky

  2. Introduction • 2002 – Blum, Gnutzman and Smilansky: a “Chaotic billiards can be discerned by counting nodal domains.”

  3. Research goal – Characterizing billiards by investigating geometrical features of the nodal domains: • Helmholtz equation on 2d surface (Dirichlet Boundary conditions): - the total number of nodal domains of .

  4. Consider the dimensionless parameter:

  5. Consider the dimensionless parameter:

  6. For an energy interval: , define a distribution function:

  7. Is there a limiting distribution? • What can we tell about the distribution?

  8. Rectangle

  9. Rectangle

  10. Rectangle

  11. Rectangle • Compact support: 2. Continuous and differentiable 3. 4.

  12. Rectangle • the geometry of the wave function is determined by the energy partition between the two degrees of freedom.

  13. Rectangle • can be determined by the classical trajectory alone. Action-angle variables:

  14. Disc • the nodal lines were estimated using SC method, neglecting terms of order .

  15. n’=1 n’=2 n’=3 n’=4

  16. Same universal features for the two surfaces: Rectangle Disc

  17. Disc • Compact support: 2. Continuous and differentiable 3. 4. m=o n=1

  18. Surfaces of revolution • Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve). • Same approximations were taken as for the Disc.

  19. Surfaces of revolution • Simple surfaces of revolution were investigated (Smooth everywhere, single maxima of the profile curve). • Same approximations were taken as for the Disc.

  20. n’=4 n’=3 n’=2 n’=1

  21. For the Disc: • For a surface of revolution:

  22. Following those notations:

  23. “Classical Calculation”:

  24. “Classical Calculation”: • Look at • (Classical • Trajectory)

  25. “Classical Calculation”: 2. Find a point along the trajectory for which:

  26. “Classical Calculation”: 3. Calculate

  27. Separable surfaces • In the SC limit, has a smooth limiting distribution with the universal characteristics: - Same compact support:- diverge like at the lower support- go to finite positive value at the upper support 2. can be deduced (in the SC limit) knowing the classical trajectory solely.

  28. Random waves

  29. Random waves Two properties of the Nodal Domains were investigated: 1.Geometrical: 2. Topological: genus – or: how many holes? G=0 G=1 G=2

  30. Random waves

  31. Random waves

  32. Random waves

  33. Random waves

  34. Random waves

  35. Random waves

  36. Random waves Model: ellipses with equally distributed eccentricity and area in the interval:

  37. Random waves d

  38. Genus The genus distributes as a power law!

  39. Genus In order to find a limiting power law – check it on the sphere

  40. Genus Power law? Saturation? Fisher’s exp:

  41. Random waves • The distribution function has different features for separable billiards and for random waves. • The topological structure (i.e. genus distribution) shows complicate behavior – decays (at most) as a power-low.

  42. Open questions: • Connection between classical Trajectories and . • Analytic derivation of for random waves. • Statistical derivation of the genus distribution • Chaotic billiards.

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