格点上两粒子薛定谔算符的阈值效应
Threshold effects of the two-particle Schrödinger operators on lattices
摘要：
我们考虑了$d$维环面$\mathbb{T}^d$中的一类具有固定的二粒子准动量$k$的二粒子薛定谔算符$H_μ(k)=H_{0}(k)+μV, \,μ>0,$，它与$d$维超立方晶格$\mathbb{Z}% ^d$上的两个全同量子力学粒子（玻色子）通过短程对势相互作用的玻色-哈伯德哈密顿量$H_μ$有关。本文研究了在本质谱阈值以下，依赖于粒子的相互作用能$μ>0$和准动量$k\in \mathbb{T}^d$的$H_μ(k)$本征值的存在性.我们证明了阈值（本质谱的底部），作为一个奇点（阈值共振或阈值本征值），创建本征值低于本质谱下的耦合常数$μ>0$和准动量$k$的粒子的扰动。此外，我们表明，如果阈值是一个定期点，那么它不会创建任何特征值下的耦合常数$μ>0$和准动量$k$的小扰动。
英文摘要：
We consider a wide class of the two-particle Schrödinger operators $H_μ(k)=H_{0}(k)+μV, \,μ>0,$ with a fixed two-particle quasi-momentum $k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the Bose-Hubbard hamiltonian $H_μ$ of a system of two identical quantum-mechanical particles (bosons) on the $d$- dimensional hypercubic lattice $\mathbb{Z}% ^d$ interacting via short-range pair potentials. We study the existence of eigenvalues of $H_μ(k)$ below the threshold of the essential spectrum depending on the interaction energy $μ>0$ and the quasi-momentum $k\in \mathbb{T}^d$ of particles. We prove that the threshold (bottom of the essential spectrum), as a singular point (a threshold resonance or a threshold eigenvalue), creates eigenvalues below the essential spectrum under perturbations of both the coupling constant $μ>0$ and the quasi-momentum $k$ of the particles. Moreover, we show that if the threshold is a regular point, then it does not create any eigenvalues under small perturbations of the coupling constant $μ>0$ and the quasi-momentum $k$.
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