The author considers, in the space , the following problem
with boundary conditions
and with the jump conditions
where and are real, , , , .
The author studies only the case
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 and the number are real due to the condition (4) the eigenfunctions are orthogonal.
If, , denotes the eigenvalues of the problem (1)–(3), then
where the values correspond to the case , i.e. are the roots of the equations
Note that .
The proof is based on the transformation operators (the author uses the methods of 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.