Asymptotic behavior of eigenvalues of Schr\"odinger type operators with degenerate kinetic energy

We study the eigenvalues of Schr\"odinger type operators $T + \lambda V$ and their asymptotic behavior in the small coupling limit $\lambda \to 0$, in the case where the symbol of the kinetic energy, $T(p)$, strongly degenerates on a non-trivial manifold of codimension one.