2
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these eigenvalues should be separated when sorted in the correct way but it is not the case when passing to Plot3D
Plot3D[Evaluate[Sort[Eigenvalues[{{2 Sin[\[Pi]/6-y],-1,-E^(3 I x)},{-1,2 Sin[\[Pi]/6+y],-1},{-E^(-3 I x),-1,-2 Cos[y]}}]]],{x,-\[Pi]/3,\[Pi]/3},{y,-\[Pi],\[Pi]},BoxRatios->{1,1,1},Mesh-