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On the asymptotics of the spectral function of a one-dimensional Dirac system on a closed interval. (Russian) Zbl 0608.34024

By the method of V. A. Sadovnichij [Differ. Uravn. 10, 1808-1818 (1974; Zbl 0311.34028)] the author considers the system \[ (1)\quad u'- (a(x)+\lambda)v=0,\quad v'+(b(x)+\lambda)u=0 \] with boundary conditions \((2)\quad u(0)\cos \alpha +v(0)\sin \alpha =0,\) \(u(\pi)\cos \beta +v(\pi)\sin \beta =0,\) where \(a,b\in C^ 1([0,\pi])\) real-valued functions, \(\alpha\),\(\beta\),\(\lambda\in R\). Let \(\{\lambda_ n\}\) be the sequence of eigenvalues of problem (1), (2) and \(\{(u_ n,v_ n)\}\) the corresponding sequence of the proper vector functions, \(u_ n(0)=-\sin \alpha\), \(v_ n(0)=\cos \alpha\). The function \[ \rho (\lambda)=- \sum_{\lambda <\lambda_ n<0}\frac{1}{\gamma_ n},\quad \lambda <0,\quad \sum_{0\leq \lambda_ n<\lambda}\frac{1}{\gamma_ n},\quad \lambda \geq 0, \] where \(\gamma_ n=\int^{\pi}_{0}(u^ 2_ n(x)+v^ 2_ n(x))dx\) is called the spectral function of problem (1), (2). The paper presents the asymptotic behaviour of the spectral function \(\rho\) (\(\lambda)\).
Reviewer: S.Staněk

MSC:

34L99 Ordinary differential operators
34B05 Linear boundary value problems for ordinary differential equations
33E99 Other special functions

Citations:

Zbl 0311.34028
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