Abstract
In this paper, we study the regularized trace of a two-fold differentiation operator with non-local matching conditions on a star graph consisting of arcs of the same length. We consider both the integrable case, when the potentials belong to the space L_{1}, and the singular case, in which the potentials admit more general features, including distributions. The main attention is paid to the derivation of the asymptotic decomposition of the characteristic function corresponding to the boundary value problem on a graph and the calculation of regularized traces using spectral theory methods. The main goal is to calculate the first regularized trace of an operator, which is defined as the limit of the sum of the differences of the eigenvalues of the operator and its modification. It is shown that in the integrable case, the regularized trace is a linear functional of the potential coefficients, whereas in the singular case (when the potentials are represented as generalized functions), it acquires a nonlinear dependence. Explicit formulas for the regularized trace using characteristic determinants and integral representation methods are derived. The results of this work generalize the well-known formulas of regularized traces applied to operators on a segment to the case of more complex structures such as graphs. The work is of interest to specialists in the field of spectral theory of operators and differential equations on graphs.