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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 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 The geometrical contents of the spectrum
The sequenceofnodal counts n =20 n=8 Sturm (1836) : For d=1 : n = n Courant (1923) : For d>1 : n n
Counting Nodal Domains: Separable systems Rectangle, Disc “billiards” in R2 Surfaces of revolution Liouville surfaces Main Feature – Checkerboard structure
Simple Surfaces of Revolution (SSR) for a few ellipsoids n(m) simple surfaces: n’’(m) 0 m
Bohr Sommerfeld (EBK) quantization
Nodal counting Order the spectrum using the spectral counting function: The nodal count sequence : The cumulative nodal count:
Cmod(k) C(k)
A trace formula for the nodal sequence Cumulative nodal counting
Numerical simulation: the smooth term Ellipsoid of revolution (c(k) – a k2)/k2 c(k)~a k2 k k k
The fluctuating part = c(k) - smooth (k) Correct power-law
The scaled fluctuating part: Its Fourier transform = the spectrum of periodic orbits lengths
The main steps in the derivation Poisson summation Semi-classical (EBK) n+1/2 ! n
Change of variables: Approximate: Integration limit: Another change of variables
The oscillatory term Saddle point integration: Picks up periodic tori with action: Collecting the terms one gets the trace formula
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
Gnutzmann films presents Nodal domains are created or merged by fission or fusion