Eigenvalues of periodic Sturm Liouville problems

For h-periodic coefficients and any integer k>2 it is well known that the eigenvalues of some self-adjoint complex boundary condition on the interval [a,a+h] are the same as the periodic eigenvalues on the interval [a,a+kh]. For each k we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. In addition, we prove an analogous result for semi-periodic eigenvalues.