Estimates of Periodic Potentials in Terms of Gap Lengths

Abstract:

Define the Hill operator T≡−d ²/dx ²+q(x) in L ²(ℝ) and suppose q∈L ²(0, 1) is a 1-periodic real potential, ∫₀ ¹ q(x)dx≡ 0. We prove the estimate ∥γ∥≥ 2 ∥γ∥(1+∥γ∥^1/3), where ∥γ∥²=∑ _{n ≥ 1}|γ _n |² and |γ _n |≥ 0, n≥ 1, is the gap length of T.

Dedicated to my teacher Mikhail Birman on the occasion of his 70th birthday