On the study of the spectrum of a functional-differential operator with a summable potential

The paper deals with a functional-differential operator of the eighth order with a summable potential. The boundary conditions are separated. Functional-differential operators of this kind arise in the study of vibrations of beams and bridges made up of materials of different density. To solve the functional-differential equation that defines a differential operator, the method of variation of constants is applied. The solution of the initial functional-differential equation is reduced to the solution of the Volterra integral equation. The resulting Volterra integral equation is solved by Picard's method of successive approximations. As a result of the investigation of the integral equation, asymptotic formulas and estimates for the solutions of the functional-differential equation that defines the differential operator are obtained. For large values of the spectral parameter, the asymptotics of the solutions of the differential equation defining the differential operator is derived. Similar to the asymptotic estimates of solutions of the differential operator of the second order with smooth and piecewise smooth coefficients, asymptotic estimates of solutions of the initial functional differential equation are established. The obtained asymptotic formulas are used to study the boundary conditions. As a result, we come to the study of the roots of a function represented as a determinant of the eighth order. To find the roots of this function, it is necessary to study the indicator diagram. The roots of the eigenvalue equation are in eight sectors of an infinitesimal solution, defined by the indicator diagram. The behavior of the roots of this equation in each of the sectors of the indicator diagram and the asymptotics of the eigenvalues of the differential operator under study are studied.