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

The sum of the eigenvalues {λn} of an operator is usually called its trace. For the eigenvalues λn of an differential operator, the series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum_n \lambda_n}$$\end{document} , generally speaking, diverges; however, it can be regularized by subtracting from λn 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 ≥ 2. Regularized trace formula of this operator is established with residue techniques in complex analysis.