In this paper is studied the spectral problem for a discontinuous second order differential operator with a summabl potential function and a spectral parameter in conjugation conditions, that arises by solving the problem on vibrations of a loaded string with free ends. In the case of a summable potential function, using abstract theorems on the stability of basis properties of multiple systems in a Banach space with respect to certain transformations, as well as theorems on the completeness and minimality of the system of eigenfunctions of the spectral problem in the spaces and