On Sturm-Liouville operators with discontinuity conditions inside an interval (Q820008)

The author considers, in the space \(L_2(0,\pi)\), the following problem \[ -y''+q(x)y=k^2y,\quad 0<x<\pi,\tag{1} \] with boundary conditions \[ y'(0)=0,\quad y(\pi)=0,\tag{2} \] and with the jump conditions \[ y(d+0)=ay(d-0),\quad y'(d+0)=by'(d-0), \tag{3} \] where \(q(x)\) and \(a\) are real, \(d\in(\frac{\pi}{2},\pi)\), \(a>0\), \(a\neq1\), \(q\in L_2(0,\pi)\). The author studies only the case \[ b=a^{-1}\tag{4} \] for the jump condition (3). Some references about mechanics, physics, etc., problems which generate boundary-value problems with discontinuities inside the interval are given. As the potential \(q(x)\) and the number \(a\) are real due to the condition (4) the eigenfunctions are orthogonal. If, \(k=1,2,\dots\), \(\lambda_n=k^2_n\) denotes the eigenvalues of the problem (1)--(3), then \[ k_n=k_n^o+\frac{c_n}{k_n^o},\quad c_n=O(1),\quad n\to\infty, \] where the values \(k_n^o\) correspond to the case \(q(x)\equiv0\), i.e. \(k_n^o\) are the roots of the equations \[ \left(a+\frac1{a}\right)\cos k\pi+\left(a-\frac1{a}\right)\cos k(2d-\pi)=0. \] Note that \(\inf_{n,m}|k_n^o-k_m^o|>0\). The proof is based on the transformation operators (the author uses the methods of \textit{V. A. Marchenko} [Sturm-Liouville operators and applications, Translated from the Russian by A. Iacob, Basel: Birkhäuser (1986; Zbl 0592.34011)]. Later the author proves the uniqueness for the inverse problems: the reconstruction of the boundary-value problem (1)--(3) from the Weyl function, from spectral data and from two spectra.