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

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Abstract

The sum of the eigenvalues {λ n } of an operator is usually called its trace. For the eigenvalues λ n of an differential operator, the series \({\sum_n \lambda_n}\) , generally speaking, diverges; however, it can be regularized by subtracting from λ 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 ≥ 2. Regularized trace formula of this operator is established with residue techniques in complex analysis.

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Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, People’s Republic of China

    Chuan-Fu Yang

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  1. Chuan-Fu Yang
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Correspondence to Chuan-Fu Yang.

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Communicated by Nobuaki Obata.

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Yang, CF. Regularized Trace for Sturm–Liouville Differential Operator on a Star-Shaped Graph. Complex Anal. Oper. Theory 7, 1185–1196 (2013). https://doi.org/10.1007/s11785-011-0193-7

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  • DOI: https://doi.org/10.1007/s11785-011-0193-7

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