38、Spectral Gap and Dirichlet Ground State Analysis

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于 2025-08-22 14:17:29 发布

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分类专栏：图的谱几何：连接数学与现实文章标签： spectral gap Dirichlet eigenvalue symmetrisation technique

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Spectral Gap and Dirichlet Ground State Analysis

1. Introduction to Spectral Gap Concepts

In the study of graphs, understanding the spectral gap is crucial. For a loop, the first eigenvalue is doubly degenerate, while for an interval, it is simple. The cycle (S_{2L}) can be derived from (\Gamma^2) by converting its vertices into degree - two vertices. This method of obtaining spectral inequalities is known as graph surgery.

2. Symmetrisation Technique

2.1 Main Idea

The symmetrisation technique aims to introduce a special transformation that maps functions from (L^2(\Gamma)) to functions from (L^2[0, L]). This transformation helps in comparing the eigenvalues of the standard Laplacians (L^{st}(\Gamma)) and (L^{st}([0, L])). The s