Asymptotics of eigenstates of elliptic problems with mixed boundary data   on domains tending to infinity
  We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions. 
Michel Chipot, Prosenjit Roy, Itai Shafrir
math.AP