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}}}}]