Delio's Third Example
Consider a lollipop graph where the stick has fixed length 2 and the length of the loop takes length L. Then as L increases, the nth eigenvalue increases continuously toward zero, except for the ground state eigenvalue which is identically zero.
First we loop over L, and save the same number of eigenvalues for each value. Also increase the number of discretization points.
Setup the data structures
numEigs=6;
Lmin = 3; Lmax = 10;
L2 = linspace(Lmin,Lmax,10*(Lmax-Lmin)+1);
eigMatrix = zeros(numEigs,numLengths+1);
Plot the layout of the quantum graph
Phi1 = quantumGraphFromTemplate('multibell',nBells=1,LVec=[2;L2(1)],discretization='Chebyshev');
Phi1.plot('layout')
Calculate the eigenvalues for changing L
for k=1:numLengths+1
LL = [2;L2(k)];
Phi1 = quantumGraphFromTemplate('multibell',nBells=1,LVec=LL,discretization='Chebyshev',nx=[32; 128]);
[V1,d1]=Phi1.eigs(numEigs);
eigMatrix(:,k)=d1;
end
Plot the Eigenvalues
plot(L2,eigMatrix,'color',mcolor('b'))
xlabel('$L$')
ylabel('$\omega_k$')
xline(4)
xline(8)
axis([Lmin Lmax min(eigMatrix(:)) 0.5])
At 
and 
, marked by vertical lines, there is a resonance between the lengths of the two segments, leading to a multiplicity-two eigenvalue, so two of the curves are tangent.
and , marked by vertical lines, there is a resonance between the lengths of the two segments, leading to a multiplicity-two eigenvalue, so two of the curves are tangent.