We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of \({\mathbb {R}}^2\). Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegő type inequality as well as a new reformulation of a Faber–Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber–Krahn type inequality.

中文翻译：

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有界域中狄拉克算子的变分公式。谱几何不等式的应用

我们在\({\mathbb {R}}^2\)的光滑有界域中研究了具有无限质量边界条件的狄拉克算子的光谱特征。受光谱几何不等式的启发，我们证明了一个非线性变分公式来表征其主要特征值。事实证明，这种表征非常稳健，并允许简单证明 Szegő 型不等式以及该算子的 Faber-Krahn 型不等式的新重新表述。该论文补充了强有力的数值证据，支持 Faber-Krahn 型不等式的存在。