Pseudospectra of a Spectral Approximation of an Anharmonic Oscillator

This figure illustrates pseudospectra of an anharmonic oscillator, -u_xx + cx²u, for c=1+3i. The figure above was obtained from a spectral approximation of dimension N=69. Since this discretization includes numerous spurious eigenvalues, the matrix was then projected onto an appropriate invariant subspace to accelerate computation of the pseudospectra. This example was first studied by Davies [Dav99a], [Dav00b].

Use the following MATLAB code compute a similar image using EigTool.

  N = 70;
  [D,x] = cheb(N); % cheb.m from Trefethen's "Spectral Methods in MATLAB"
  x = x(2:N);  
  L = 6; x = L*x; D = D/L;                   % rescale to [-L,L]
  A = -D^2; A = A(2:N,2:N) + (1+3i)*diag(x.^2);
  opts.npts=50;
  opts.ax = [0 50 0 40];
  opts.levels = -10:.5:0;
  opts.proj_lev = .7;
  eigtool(A,opts)