The eigenvalue problem $\frac{d^2u}{dx^2}-6\sin(x)^2=\mu u(x)$ gives the first four eigenvalues which are 2.06, 2.26, 5.16, 6.81, and 7.74. And those eigenvalue I have calculated, but the corresponding eigenvector that I got did not agree with the Ref. https://journals.aps.org/pre/pdf/10.1103/PhysRevE.75.056602 in Fig.2. In this paper the author tell us the vector at the edge is all real, but I calculate it with complex.
Bloch wave
PRE75.056602
PRE75.056602
Clear["Global`*"]
Lx=\[Pi];
k0x=2 \[Pi]/Lx;
V=6 Sin[x]^2;
kx=0;
{eig0,funs}=NDEigensystem[{-D[u[x],{x,2}]-2I kx D[u[x],x]+(kx^2+V)u[x],u[0]== u[\[Pi]]},u[x],{x,0,\[Pi]},3,Method->{"VectorNormalization"->True,"PDEDiscretization"->{"FiniteElement",{"MeshOptions"->{"MaxCellMeasure"->0.01}}}}]
{2.06318, 6.81429, 7.74678},
kx=1;{eig2,fun2}=NDEigensystem[{-D[u[x],{x,2}]-2I kx D[u[x],x]+(kx^2+V)u[x],u[0]== u[\[Pi]]},u[x],{x,0,\[Pi]},3,Method->{"PDEDiscretization"->{"FiniteElement",{"MeshOptions"->{"MaxCellMeasure"->0.01}}}}]
{2.26673, 5.16594, 12.0926}
What's wrong? thank you!