Let q(t) be a T -periodic potential such that ∫ T 0 q(t) dt < 0. The classical Lyapunov criterion to stability of Hill’s equation −ẍ + q(t)x = 0 is ‖q−‖1 = ∫ T 0 |q−(t)|dt ≤ 4/T , where q− is the negative part of q. In this paper, we will use a relation between the (anti-)periodic and the Dirichlet eigenvalues to establish some lower bounds for the first anti-periodic eigenvalue. As a result, we will find the best Lyapunov-type stability criterion using Lα norms of q−, 1 ≤ α ≤ ∞. The numerical… CONTINUE READING