Аннотация
The paper investigates a second-order differential operator on a star graph. A special class of differential operators on a star graph with simple eigenvalues is chosen. The structure of the characteristic determinants of such operators is studied. In the case of the Sturm-Liouville operator with constant coefficients, the formula of the first regularized trace is written out. The main purpose of this work is to calculate the regularized trace of an operator on a star graph, which differs from calculations for similar operators on segments considered in other papers. The article also describes in detail the properties of the eigenvalues of an operator, including the theorem that the eigenvalues of an operator coincide with the zeros of an entire function, and the algebraic multiplicity of each eigenvalue is equal to the multiplicity of zero of the function. For clarity, the results of the work are presented through the characteristic determinant of the operator and numerical series that describe the behavior of the regularized trace. Using the methods of function theory and analytical series, the first regularized trace is calculated, which is an important step in studying the spectral characteristics of an operator on graphs. The article is of interest to specialists in the theory of spectral operators and differential equations on graphs, as well as to researchers involved in calculating operator traces and analyzing their asymptotic properties.