AI 中文总结
本文研究耦合于大质量标量玻色场的软量子波导的Nelson型哈密顿量，通过对比弯曲与直形波导的哈密顿量，分析其谱性质并推导弯曲情形下离散谱存在的充分条件。
AI 中文摘要
我们研究了一种Nelson型哈密顿量，它描述了在平面软量子波导中运动的大质量无自旋非相对论粒子，且该粒子与大质量标量玻色场线性耦合。我们的谱分析依赖于将弯曲波导与其直形对应物进行比较。对于直形哈密顿量，纵向平移不变性产生纤维分解，我们证明其谱完全是本质的且无间隙。对于弯曲哈密顿量，我们建立了本质谱底部的HVZ型公式，该公式涉及直形系统的本质谱和单玻色阈值。我们还推导了弯曲情形下存在非空离散谱的充分条件，该条件以有效一维薛定谔算子的形式表述。
英文摘要
We study a Nelson-type Hamiltonian describing a massive spinless non-relativistic particle moving in a planar soft quantum waveguide and linearly coupled to a massive scalar bosonic field. Our spectral analysis relies on comparing the curved waveguide with its straight counterpart. For the straight Hamiltonian, longitudinal translation invariance yields a fiber decomposition, and we prove that its spectrum is purely essential and gapless. For the curved Hamiltonian, we establish an HVZ-type formula for the bottom of the essential spectrum, involving that of the straight system and the one-boson threshold. We also derive a sufficient condition for the existence of non-empty discrete spectrum in the curved case, expressed in terms of an effective one-dimensional Schrödinger operator.
