Regularized Trace for Sturm-Liouville Differential Operator on a Star-Shaped Graph

The sum of the eigenvalues {lambda (n) } of an operator is usually called its trace. For the eigenvalues lambda (n) of an differential operator, the series , generally speaking, diverges; however, it can be regularized by subtracting from lambda (n) the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace. In this work, we consider the spectral problem for Sturm-Liouville differential operator on d-star-type graph with a Kirchhoff-type condition in the internal vertex, where the integer d a parts per thousand yen 2. Regularized trace formula of this operator is established with residue techniques in complex analysis.