Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator
A coupled harmonic oscillator with quadratic coupling - $$ H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2, $$ is known to be non-integrable, hence chaotic (for reference look at Quantum-chaos, <PERSON>). That means if I find the neighborhood level spacing ($ e_i - e_{i-1} $) distribution, I should be able to see Wigner-Dyson (WD) distribution.
To find the neighborhood level spacing distribution (NLSD), I need to find eigenvalues of the Hamiltonian. 768
As the underlying Hilbert space, in this case, is infinite-dimensional, one usually finds a given number of eigenvalues, and plots the NLSD, in principle as the number of states used to find NLSD is increased, the distribution tends more toward the WD. Is there a way to estimate how many states I need to get a distribution approaching the WD? My naive approach is given below.
As I am not sure about how many states to take, I am trying a brute force method with an increasing number of states used in the calculation of how the NLSD approaches WD and how the mean level spacing ratio approaches 0.53.
I am numerically solving the time-independent <PERSON>'s equation over a box of dimension $20\times20$,
$$ -\nabla^2 \psi_n(x,y) + \left( \frac{1}{2}(x^2 + y^2) + g x^2y^2 \right)\psi_n(x,y) = E_n \psi_n(x,y), $$ subjected to the Dirichlet boundary condition $\psi_n(x,y)|_{\text{boundary}} = 0$ and finding a number of lowest (100 in this case) eigen-states in Mathematica$^®$
{eval, evec} = NDEigensystem[{ -1/2 (D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}]) + 1/2 (x^2 + y^2) u[x, y] + x^2 y^2 u[x, y], DirichletCondition[u[x, y] == 0, True]}, u[x, y], Element[{x, y}, Rectangle[{-20, -20}, {20, 20}]], 100, Method -> {"PDEDiscretization" -> {"FiniteElement", "MeshOptions" -> {"MaxCellMeasure" -> 0.05}}}];
Then the NLSD is obtained by the histogram of eval[[2 ;; -1]] - eval[[1 ;; -2]] Histogram[eval[[2 ;; -1]] - eval[[1 ;; -2]]]
This doesn't look like WD at all and has a mean level spacing ration $\left \approx 0.87$ which is also not correct. There can be two problems, 1) the number of states that I am using is too low, and 2) probably I should look into a parity sector where all the states are uncorrelated other than looking at the whole Hilbert space wh to get WD.
If there is an estimation of how many states to take to get a good approximation of WD, probably some completeness argument, i.e., how many states do you need to get a good approximation of completeness relation that might be helpful here. 935