论文标题
两粒子Schrödinger操作员对晶格的阈值效果
Threshold effects of the two-particle Schrödinger operators on lattices
论文作者
论文摘要
我们考虑一类宽类的两粒子Schrödinger操作员$h_μ（k）= h_ {0}（k）+μv，\，μ> 0，$，带有固定的两粒子quasi-momentum $ k $ $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ \ mathbb { $ d $ - 尺寸超顺从晶格$ \ mathbb {z}％ ^d $通过短距离对的电势相互作用上的相同的量子机电颗粒（玻色子）。我们研究了$H_μ（k）$的特征值的存在，这取决于相互作用的能量$μ> 0 $和quasi-momentum $ k \ in \ mathbb {t}^d $的粒子粒子。我们证明，阈值（基本频谱的底部）是一个单数点（阈值共振或阈值特征值），在耦合常数$μ> 0 $和Quasi-Momentum $ k $ k $ k $ k $ k $ k $ k $ k $ n contustrations contustrations conders contuctrum contuctrumation contuctruma的基本频谱下产生特征值。此外，我们表明，如果阈值是常规点，那么它不会在耦合常数$μ> 0 $和Quasi-momentum $ k $的小扰动下创建任何特征值。
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 $k$.
