The formula of the first reqularized trace for a differential operator with a discontinuous weight function

The author study the spectral properties of an eighth-order differential operator with a piecewise-smooth potential and a discontinuous weight function. For large values of the spectral parameter, the asymptotics of solutions of differential equations defining the operator under study is studied. With the help of the obtained asymptotics, the conditions of “conjugation” at the point of discontinuity of the coefficients, the necessity of which follows from physical considerations, are studied. The separated boundary conditions that define the operator are studied. An indicator diagram of an equation whose roots are the eigenvalues of the operator is investigated. The asymptotics of the eigenvalues of the differential operator under study is found. Using the Lidskyi — Sadovnichyi method, the first regularized trace of the differential operator is calculated.