Abstract In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; Then, we give a concrete description of each eigenspace in terms of Hermite and complex Hermite polynomials. In particular, an explicit description of the L2-holomorphic sections on the above line bundle is presented as the eigenspace of the magnetic Laplacian corresponding to the least eigenvalue. Also by using the periodization principle, the associated invariant integral operators are discussed.

中文翻译：

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关于拟环上 theta 线丛的 Bochner Laplacian 算子

摘要 在本文中，我们考虑了拟环面上theta 线丛上的Laplcian 算子，称为Bochner Laplacian。这个算子作为一个磁性拉普拉斯算子有一个规范的实现，它作用于满足函数方程的复值函数。我们研究了这种拉普拉斯算子的光谱特性，并表明它的光谱被简化为特征值 πm；然后，我们根据 Hermite 和复 Hermite 多项式对每个特征空间进行了具体描述。特别地，上述线丛上 L2 全纯截面的明确描述被表示为对应于最小特征值的磁性拉普拉斯算子的特征空间。还通过使用周期化原理，讨论了相关的不变积分算子。