Journal of Mathematical Sciences, Vol. 134, No. 4, 2006 THE INVERSE PROBLEM FOR A DISCRETE PERIODIC SCHR ¨ ODINGER OPERATOR E. Korotyaev ∗ and A. Kutsenko † UDC 517.5 We study isospectral sets for a discrete 1D Schr¨odinger operator on Z with an (N + 1)-periodic potential. We show that for small odd potentials, the isospectral set consists of 2 (N+1)/2 elements, while for large potentials, the isospectral set consists of (N + 1)! elements. Moreover, asymptotics for endpoints of the spectrum of the Schr¨ odinger operator for small (and large) potentials are determined. Bibliography: 5 titles. 1.Introduction In this paper, we study isospectral sets for a discrete (N + 1)-periodic Schr¨ odinger operator. For small odd potentials, isospectral sets consist, in principle, of 2 N+1 2 elements, while for large potentials such sets consist of (N + 1)! elements. Moreover, we describe asymptotics of boundary zones of the spectrum for the Schr¨ odinger operator for the cases of potentials with large norms and of odd potentials with small norms. Consider a periodic Schr¨ odinger operator (Ly) n = y n−1 + y n+1 + q n y n , n ∈ Z, y ∈ l 2 (Z). The potential {q n } ∞ n=−∞ is a real sequence of period N + 1, q n+N+1 = q n , n ∈ Z. We consider the following space of potentials: q ≡{q n } N+1 1 ∈Q≡  q ∈ R N+1 : N+1  1 q n =0  , ‖q‖ 2 = N+1  n=1 q 2 n . (1.1) It is well known that the spectrum of the operator L is absolutely continuous and consists of N + 1 zones σ n = σ n (q)=[λ + n ,λ − n+1 ], n =0, 1,...,N . Here λ ± n = λ ± n (q) are the endpoints of zones, and λ + 0 <λ − 1 ≤ λ + 1 < ...<λ − N ≤ λ + N <λ − N+1 . The zones are separated by the lacunas γ n = γ n (q)=(λ − n ,λ + n ) of lengths |γ n |≥ 0. It is possible that a lacuna is degenerate, i.e., that |γ n | = 0. Denote by ϕ n (λ, q) and ϑ n (λ, q), n ∈ Z, fundamental solutions of the equation y n−1 + y n+1 + q n y n = λy n , λ ∈ C,n ∈ Z, (1.2) with initial values ϕ 0 (λ, q) ≡ ϑ 1 (λ, q) ≡ 0 and ϕ 1 (λ, q) ≡ ϑ 0 (λ, q) ≡ 1, respectively. The function ∆(λ, q)= ϕ N+2 (λ, q)+ ϑ N+1 (λ, q) is called the Lyapunov function for the operator L. We see that ∆,ϕ n , and ϑ n ,n ≥ 1, are polynomials in (λ, q) ∈ C N+1 . The spectrum of the operator L is given by the formula σ(q)= {λ ∈ R : |∆(λ, q)|≤ 2}. Note that (−1) N+1−n ∆(λ ± n ,q) = 2, n =0,...,N + 1. We introduce the following parameters which are convenient for the description of the spectrum for various potentials. For any n =1,...,N , there exists a point λ n = λ n (q) ∈ [λ − n ,λ + n ] such that ∆ ′ (λ n ,q)=0, ∆ ′′ (λ n ,q) =0, and (−1) N+1−n ∆(λ n ,q) ≥ 2. (1.3) Here and below, we use the notation ( ′ )= ∂/∂λ. Define a mapping h : Q→ R N , h(q)= {h n (q)} N 1 , by the formula ∆(λ n ,q) = 2(−1) N+1−n cosh h n , h n ≥ 0. (1.4) ∗ Institut f¨ ur Mathematik, Humboldt Universit¨at zu Berlin, Germany, e-mail: ek@mathematik.hu-berlin.de. † St.Petersburg State University, St.Petersburg, Russia, Institut f¨ ur Mathematik, Universit¨at Potsdam, Germany, e-mail: kucenko@math.uni-potsdam.de. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 96–101. Original article submitted April 20, 2004. 2292 1072-3374/06/1344-2292 c 2006 Springer Science+Business Media, Inc.