On the study of spectral properties of differential operators of even order with discontinuous weight function

The boundary value problem for a differential operator of high even order, whose coefficients are discontinuous functions at some interior point of the segment on which the operator is considered, is studied. At the point of discontinuity of the coefficients, certain conditions of «conjugation» that follow from the physical conditions are required. The boundary conditions of the considered boundary value problem are separated and depend on several parameters. Thus simultaneously the spectral properties of a family of differential operators are studied. The weight function of the operator is piecewise constant on the interval of the definition of the differential operator. For large values of the spectral parameter, the asymptotics of the solutions of the differential equations determining the operator under investigation is derived. Using this asymptotics, the conditions of “conjugation” are studied. The obtained formulas allow to investigate the boundary conditions of the considered boundary value problem. As a result, we have derived an equation for the eigenvalues of the studied operator. It is proved that the eigenvalues of the operator are the roots of some entire function. The indicator diagram of the equation for the eigenvalues of the operator is studied. It is proved that the spectrum of the operator is discrete. In different sectors of the indicator diagram, the asymptotics of the eigenvalues of the studied operator is found, depending on the parameters of the boundary conditions. The found formulas allow us to find the asymptotics of the eigenfunctions of the operator and to calculate the regularized traces of this operator.