Abstract:
For some class of boundary conditions, generated Sturm–Lioville operator, we prove existence and uniqueness of solution of corresponding Gelfand–Levitan equation. In second part we prove that if $\{\lambda_n \}_{n \in \mathbb{Z}}$ is the set of eigenvalues of selfadjoint Dirac operator on $(0, \pi)$, then the system of vector-functions \begin{equation*} \bigg\{ \bigg( \begin{matrix} \sin \lambda_n x \\ -\cos \lambda_n x \end{matrix} \bigg) \bigg\}_{n \in \mathbb{Z}} \end{equation*} is a Riesz bases in Hilbert space $L^2 ([0, \pi], \mathbb{C}^2)$.
Key words and phrases:inverse problem, Gelfand–Levitan equations, uniqueness of solution, Riesz bases.
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