On the studying the spectrum of differential operators' family whose potentials converge to the Dirac delta function

Background. The paper proposes a new method for studying differential operators with discontinuous coefficients. We study a sequence of differential operators of high even order whose potentials converge to the Dirac delta function. It is assumed that the operator's potential is a piecewise-summable function on the segment of the operator task. At the points of discontinuity of the potential, the fulfillment of the “gluing” conditions is required to correctly determine the solutions of the corresponding differential equations. The spectral properties of differential operators defined on a finite segment with one of the types of separated boundary conditions are investigated. For large values of the spectral parameter, the Naimark method is used to obtain the asymptotics of the fundamental system of solutions of the corresponding differential equations. With the help of this asymptotics, the conditions for “gluing” the considered differential operator are studied. Then the boundary conditions of the operator under study are studied. As a result, we derive an eigenvalue equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular hexadecimal, is investigated. In various sectors of the indicator diagram, the method of successive Picard approximations has been used to find the eigenvalue asymptotics of the studied differential operators. In the limiting case, the found asymptotics of the eigenvalues tends to the asymptotics of the eigenvalues of an operator whose potential is the Dirac delta function. Materials and methods. The asymptotics of the solutions' fundamental system of differential equations with summable potentials for large values of the spectral parameter is obtained by the generalized Naimark method. To find the roots of the equation for the eigenvalues of the operator under study, the Bellman-cook method is used to study the indicator diagram, which is a regular hexadecagon. The asymptotics of the eigenvalues of the studied differential operators in various sectors of the indicator diagram are found by the method of successive Picard approximations. Results. The spectrum of a previously unexplored family of differential operators of high even order whose potentials converge to the Dirac Delta function is studied. Taking into account the “gluing” condition at the points of discontinuity of the potentials, it is proved that the eigenvalue equation is a quasi-polynomial, the roots of which can be found by the Bellman-Cook method. Similar results can be obtained for other types of separated boundary conditions. Conclusions. The new results obtained on the asymptotics of the spectrum of a family of differential operators can be applied to the study of the basis property of the eigenfunctions of similar operators, the study of the Green's function, and the calculation of formulas for regularized traces of operators whose sequence of potentials converges to the Dirac delta function. The delta potential method is used in physics to study short-range impurities and defects in various systems. In atomic and nuclear physics, the model of point potentials is very popular; this confirms the need to study operators with delta potentials.