Regularized trace of a multipoint boundary value problem with a discontinuous weight function

The article proposes a method for calculating the regularized trace for a differential operator with a piecewise smooth potential and multipoint boundary conditions. The weight function of the differential operator is discontinuous. Using the Naimark method on the sections of the continuity of the potential and the weight function for large values of the spectral parameter, the asymptotics of solutions of differential equations defining the operator under study is obtained. The asymptotics of the solutions enables us to study the conditions of «conjugation» at the point of discontinuity of the coefficients. The necessity of the conditions of «conjugation» follows from physical considerations. The studied boundary value problems arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The multipoint boundary conditions defining the operator are studied. The technically difficult part of the study was successfully completed — the indicator diagram of the equation whose roots are the eigenvalues of the operator was studied. The asymptotics of the eigenvalues of the operator is calculated. Using the asymptotics of the eigenvalues by the Lidsky–Sadovnichy method, the first regularized trace of the differential operator is calculated.