Mathematics

CLC number: O175.9

Spectral Properties of Dirac Operator with λ-Dependent Boundary Condition

边界条件含有谱参数的Dirac算子的谱性质

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

^† Corresponding author. E-mail: dgf96520@163.com

Received: 28 November 2024

Abstract

In this study, we mainly discuss some spectral properties of the Dirac operator with eigenparameter-dependent boundary condition. Initially, we reformulate the spectral problem into linear operator eigenparameter problem in a suitable Hilbert space, and obtain some pivotal properties of self-adjoint operator. Subsequently, by establishing the boundary condition space and constructing the embedded mapping, we show that the simple eigenvalue branch of this system is not only continuous, but also smooth. We then obtain the differential expressions of the eigenvalue branch in the sense of Fréchet derivative.

摘要

本文主要研究边界条件含有谱参数的Dirac算子特征值问题的一些谱性质。首先，通过建立适当的Hilbert空间将谱问题转化为线性算子特征值问题，并推导出了自伴算子的一些重要性质。其次，通过建立边界条件空间，构造嵌入映射，证明了Dirac算子的简单特征值分支不仅是连续的，而且是光滑的。最后，在Fréchet导数意义下，我们得到了特征值分支关于所有参数的微分表达式。