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Inverse spectral results for AKNS systems with partial information on the potentials. (English) Zbl 0998.34010
Let $\{\mu_n(\alpha)\}_{n\in\bbfZ}$ be the spectrum of the boundary value problem $$ BY'+Q(x)Y=\mu Y,\quad y_2(0)=y_1(1)\cos\alpha-y_2(1)\sin\alpha=0, \alpha\in[0,\pi), $$ with $$ Y=\left( \smallmatrix y_1 \\ y_2 \endsmallmatrix\right),\quad B=\left( \smallmatrix 0 & -1 \\ 1 & 0 \endsmallmatrix\right),\quad Q(x)=\left( \smallmatrix -q(x) & p(x) \\ p(x) & q(x) \endsmallmatrix\right), $$ $\varphi:= q-ip\in L^2(0,1)$, and $q$ and $p$ are real-valued. The main result of the paper is the following uniqueness theorem: Let $a\in [0,1], \alpha\ne\beta, 1/l+1/k\ge 2a$. Then the specification of $\varphi$ on $(a,1)$ and $\{\mu_{ln}(\alpha), \mu_{kn}(\beta)\}_{n\in \bbfZ}$ uniquely determines $\alpha, \beta$ and $\varphi$ a.e. on $(0,a)$.

34A55Inverse problems of ODE
34B05Linear boundary value problems for ODE
34L40Particular ordinary differential operators
47E05Ordinary differential operators
47A50Equations and inequalities involving linear operators, with vector unknowns
47A75Eigenvalue problems (linear operators)
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