In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated with Luttinger’s model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.

中文翻译：

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热哈密顿量的谱理论：一维情况

1964 年，JM Luttinger 引入了量子热传输模型。在本文中，我们研究了与 Luttinger 模型相关的哈密顿算子的谱理论，特别关注一维情况。结果表明（所谓的）热哈密顿量具有自伴随扩展的单参数族，并且显式计算了谱、时间传播子群和格林函数。此外，分析了卷积型电位的散射。最后，也完全解决了相关的经典问题，从而提供了经典和量子行为之间的比较。本文旨在成为构建热哈密顿量完整理论的第一个贡献。