The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function p(lambda) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval p(-1)(n/2) in R yield the corresponding periodic or anti-periodic eigenvalues, However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrodinger operators with periodic potentials.