Proofs of main theorems Sample Clauses

Proofs of main theorems. We start by proving Theorem 1 for m =0 and q = 1/2 using the approach in Sec. 7. It is inessential in our argument that the potential has form (5). It suffices to assume that v(x) is a smooth periodic function with a unique point of global maximum. We again stress that the symmetry x → —x is not required. We study the case of arbitrary q later (see more details in Appendix D). We now proceed to prove the theorems in Sec. 4. For simplicity, we assume that β = 1 without loss of generality. ^ Proof of Theorem 3. We first establish the estimates and asymptotic approximations for the band widths (i.e., dwell on the case q = 1/2). For this, we consider the operator H on the cylinder Z(2). According to ▇▇▇▇▇▇▇ 6, the operator H^ |Vν is determined by the matrix P = SI + W + O(e—2σ/h) ≡ S w12 + O(e—2σ/h).