This work considers the discontinuous boundary value problem with the eigenvalue parameter appearing linearly in the boundary conditions for a second order ordinary differential equation:
where is a spectral parameter, for and for , are real numbers, and
Firstly, the given boundary value problem is reduced to an linear operator in a special Hilbert space . Then, it is shown that the operator is selfadjoint in the space . It is proved that the operator has only point spectrum and the corresponding eigenfunctions form an orthonormal basis in .
There are many studies about boundary problems with discontinuous coefficients and transmission conditions, completeness, minimality and basis property of the eigenfunctions of this boundary value problem [for example, see A. Gomilko and V. Pivovarcik, Math. Nachr. 245, 72–93 (2002; Zbl 1023.34023); V. Pivovarcik, Asymptotic Analysis., 26 (2001)]; R. Kh. Amirov, A. Z. Ozkan and B. Keskin, Integral Transforms Spec. Funct. 20, No. 7–8, 607–618 (2009; Zbl 1181.34019) and others].