On the study of the spectral properties of diﬀerential operators with a smooth weight function
In this paper we study the spectral properties of a third-order diﬀerential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying diﬀerential operators with summable potential is a development of the method of studying operators with piecewise smooth coeﬃcients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of diﬀerent densities. The diﬀerential equation deﬁning the diﬀerential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a diﬀerential equation deﬁning a diﬀerential operator. For large values of the spectral parameter, the asymptotics of solutions of the diﬀerential equation that deﬁnes the diﬀerential operator is derived. Asymptotic estimates of solutions of a diﬀerential equation are obtained in the same way as asymptotic estimates of solutions of a diﬀerential operator with smooth coeﬃcients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an inﬁnitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the diﬀerential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the diﬀerential operator under study.