The Rotation Number Approach to Eigenvalues of the One-Dimensional p-Laplacian with Periodic Potentials

The paper studies the periodic and anti-periodic eigenvaluesof the one-dimensional p-Laplacian with a periodic potential.After a rotation number function () has been introduced, itis proved that for any non-negative integer n, the endpointsof the interval ^–1(n/2) in R yield the corresponding periodicor anti-periodic eigenvalues. However, as in the Dirichlet problemof the higher dimensional p-Laplacian, it remains open if theseeigenvalues represent all periodic and anti-periodic eigenvalues.The result obtained is a partial generalization of the spectrumtheory of the one-dimensional Schrödinger operators withperiodic potentials.