p21: Eigenvalues of Mathieu operator¶

In [4]:

%config InlineBackend.figure_format='svg'
from numpy import pi,arange,sin,cos,zeros,diag,sort,real
from scipy.linalg import toeplitz
from numpy.linalg import eig
from itertools import cycle
from matplotlib.pyplot import figure,plot,xlabel,ylabel

In [5]:

N = 42; h = 2.0*pi/N; x = h*arange(1,N+1)
col = zeros(N)
col[0] = -pi**2/(3.0*h**2) - 1.0/6.0
col[1:] = -0.5*(-1.0)**arange(1,N)/sin(0.5*h*arange(1,N))**2
D2 = toeplitz(col)

ne = 11 # number of eigenvalues to plot
qq = arange(0.0, 15.0, 0.2)
data= zeros((len(qq),ne))
i = 0
for q in qq:
    evals,evecs = eig(-D2 + 2.0*q*diag(cos(2.0*x)))
    e = real(sort(evals))
    data[i,:] = e[0:ne]
    i = i + 1
    
figure(figsize=(5,10))
lines=cycle(["-","--"])
for i in range(ne):
    plot(qq,data[:,i],next(lines))
xlabel("q")
ylabel("$\\lambda$");