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On Sturm-Liouville operators with discontinuity conditions inside an interval. (English) Zbl 1168.34019
The author considers, in the space $L_2(0,\pi)$, the following problem $$ -y''+q(x)y=k^2y,\quad 0<x<\pi,\tag1 $$ with boundary conditions $$ y'(0)=0,\quad y(\pi)=0,\tag2 $$ and with the jump conditions $$ y(d+0)=ay(d-0),\quad y'(d+0)=by'(d-0), \tag3 $$ 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 $$ b=a^{-1}\tag4 $$ 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 {\it 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.
Reviewer: Evgeney V. Cheremnikh (L’viv)

34B24Sturm-Liouville theory
34L20Asymptotic distribution of eigenvalues for OD operators
Full Text: DOI
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