A note on gaps of Hill’s equation

We consider the differential operator L = − d 2 dx2 + q, q ∈ L2 = L2(S1,R) on the interval [0, 1] endowed with periodic or anti-periodic boundary conditions: y(0) = y(1), y′(0) = y′(1) or y(0) = −y(1), y′(0) = −y′(1). The corresponding differential equation −y′ ′ + qy = λy is also known as Hill’s equation with potential q. It is well known that the spectrum of L is pure point and consists of an un-bounded sequence of periodic eigenvalues λ0(q) < λ1(q) ≤ λ2(q) < λ3(q) ≤ λ4(q) <...