Traces for Sturm–Liouville Operators on a Caterpillar Graph

Abstract

In this work, we consider the spectral problems for the Sturm–Liouville operators on a caterpillar graph with the standard matching conditions in the internal vertices and the Neumann or the Dirichlet conditions in the boundary vertices. The regularized trace formulae of these operators are established by using the residue techniques of complex analysis.

Full text

Complex Analysis and Operator Theory (2024) 18:64
https://doi.org/10.1007/s11785-024-01505-5
Complex Analysis
and Operator Theory
Traces for Sturm–Liouville Operators on a Caterpillar Graph
Feng Wang1·Chuan-Fu Yang1·Natalia P. Bondarenko2
Received: 19 April 2023 / Accepted: 13 February 2024 / Published online: 25 March 2024
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024
Abstract
In this work, we consider the spectral problems for the Sturm–Liouville operators on
a caterpillar graph with the standard matching conditions in the internal vertices and
the Neumann or the Dirichlet conditions in the boundary vertices. The regularized
trace formulae of these operators are established by using the residue techniques of
complex analysis.
Keywords Sturm–Liouville operator ·Trace formula ·Caterpillar graph
Mathematics Subject Classification 34B24 ·47E05
1 Introduction
Quantum graphs refer to differential operators on metric graphs, which were intro-
duced by Ruedenberg and Scherr [24]. In recent decades, there is a lot of interest to
the study of Sturm–Liouville differential operators on graphs. On the one hand, such
problems are a natural extension of the classical Sturm–Liouville differential operators
on an interval; on the other hand, quantum graphs have many applications in natural
sciences and engineering (see [1] and the references therein).
Communicated by Petr Siegl.
BFeng Wang
wangfengmath@njust.edu.cn
Chuan-Fu Yang
chuanfuyang@njust.edu.cn
Natalia P. Bondarenko
bondarenkonp@sgu.ru
1School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing,
Jiangsu 210094, China
2S.M. Nikolskii Mathematical Institute, Peoples Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya Street, Moscow, Russian Federation 117198
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