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An inverse problem for the discrete periodic Schrödinger operator. (Russian, English) Zbl 1084.47023

Zap. Nauchn. Semin. POMI 315, 96-101 (2004); translation in J. Math. Sci., New York 134, No. 4, 2292-2294 (2006).
The authors consider the discrete \((N+1)\)-periodic Schrödinger operator \[ (Ly)_n=y_{n-1}+y_{n+1}+q_ny_n, \;\;n\in \mathbb Z, \;\;y\in l^2(\mathbb Z). \] Here \(\{q_n\}^{\infty}_{n=-\infty}\) is an \((N+1)\)-periodic sequence: \(q_{n+N+1}=q_n\), \(n\in \mathbb Z\). Consider the space of potentials \[ q\equiv \{q_n\}_1^{N+1}\in {\mathcal Q}\equiv\{q\in \mathbb R^{N+1}: \mathop{\sum} \limits_1^{N+1}q_n=0\}, \;\;\| q\| ^2=\mathop{\sum}\limits_{1}^{N+1}q_n^2. \] Let \(\{\varphi_n(\lambda)\}\) and \(\{\vartheta_n(\lambda)\}\) be fundamental solutions of the equation \[ (Ly)_n=\lambda y_n, \;\;\lambda\in \mathbb C, \;\;n\in \mathbb Z, \] which satisfy the conditions \[ \varphi_0(\lambda)\equiv\vartheta_1(\lambda)\equiv 0, \;\;\varphi_1(\lambda)\equiv\vartheta_0(\lambda)\equiv 1. \] Then the function \[ \Delta(\lambda)=\varphi_{N+2}+\vartheta_{N+1} \] is the analogue of the Lyapunov function or Hill determinant of the theory of continuous periodic Schrödinger operators. The corresponding analogue of the Marchenko–Ostrovskii mapping (see [V. A. Marchenko, “Sturm–Liouville operators and applications” (Oper. Theory, Adv. Appl. 22, Birkhäuser, Basel/Boston/Stuttgart) (1986; Zbl 0592.34011)]) is the mapping \(h:{\mathcal Q}\rightarrow \mathbb R^N, \;h(q)=\{h_n(q)\}_1^N\), defined by the formula \[ \Delta(\lambda_n,q)=2(-1)^{N+1-n}\cosh h_n, \;\;h_n\geq 0. \] The authors investigate domains of local and global isomorphism and peculiarities of the mapping \(h\).

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
39A70 Difference operators
47B39 Linear difference operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Citations:

Zbl 0592.34011
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