Asymptotics of the spectrum of family functional-differential operators with summable potential

The paper deals with a high-order functional-differential operator with a summable potential. The boundary conditions are separated and regular. Operators of this type are called loaded operators. The initial functional-differential equation is reduced to the Volterra integral equation. With the help of Picard`s method of successive approximations for large values of the spectral parameter the asymptotics formulas and estimates for solutions of the functional-differential equation that defines the differential operator are found. Eigenvalues of the studied operators are roots of the function are presented in the form of a determinant of a high order. To find the roots of this function, it is necessary to study the indicator diagram. The roots of the eigenvalue equation are found in sixteen sectors of an infinitesimal solution, determinated by the indicator diagram. The behavior of the roots of this equation is studied in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is obtained. Functional-differential operators of this kind arise when studying the vibrations of bridges and beams made up of materials of different density, with the load at an internal point.