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The inverse spectral problem for first order systems on the half line. (English) Zbl 0956.34072

Adamyan, V. M. (ed.) et al., Differential operators and related topics. Proceedings of the Mark Krein international conference on operator theory and applications, Odessa, Ukraine, August 18-22, 1997. Volume I. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 117, 199-238 (2000).
The authors study the first-order differential operator \(L\equiv B{1\over i}{d\over dx}+ Q(x)\), (on the half line \([0,\infty)\)) where \(B= \left|\begin{smallmatrix} B_1 & 0\\ 0 & B_2\end{smallmatrix}\right|\), \(B_1,B_2\in M(n,C)\) are selfadjoint positive definite matrices and \(Q: \mathbb{R}_+\to M(2n,c)\), \(\mathbb{R}_+:= [0,\infty)\), is a continuous selfadjoint off-diagonal matrix function. This operator subject to the boundary condition \(f_2(0)= Hf_1(0)\) with \(B_1= H^* B_2H\) generates a selfadjoint extension \(L_H\) of the minimal operator corresponding to \(Lf_1(0)\), \(f_2(0)\) denotes the first resp. last \(n\) components of the vector \(f(0)\).
The main purpose of the paper is to investigate the inverse spectral problem for the operator \(L_H\), i.e., to find necessary and sufficient conditions for an \(n\times n\)-matrix function \(\sigma\) to be the spectral function of the boundary value problem. The results obtained are applied to show the existence of \(2n\times 2n\) Dirac systems \((B_1= B_2= I)\) with purely absolute continuous, purely singular continuous and discrete spectrum of multiplicity \(p\), where \(1\leq p\leq n\) is arbitrary.
For the entire collection see [Zbl 1051.47001].

MSC:

34L05 General spectral theory of ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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