一维格点上费米子-玻色子对的阈值谱跃迁

中文摘要 AI 辅助

我们研究了描述一维格点$\boldsymbol{\rm Z}$上费米子-玻色子对的两粒子格点薛定谔算符，该对具有强度为$\boldsymbol{\rm μ∈R}$的零程在位相互作用，质量比为$\boldsymbol{\rm γ>0}$。利用相对坐标和关于总准动量$\boldsymbol{\rm k∈T:=(-π,π]}$的纤维分解，我们得到了$\boldsymbol{\rm ℓ^2(Z)}$中的实对称纤维哈密顿量$\boldsymbol{\rm H_{μ,γ}(k)}$，其本质谱为$\boldsymbol{\rm [2(1+γ)-2a_γ(k),\n2(1+γ)+2a_γ(k)]}$，其中$\boldsymbol{\rm a_γ(k)=\n\sqrt{1+2γ\cos k+γ^2}}$。\n当$\boldsymbol{\rm a_γ(k)>0}$且$\boldsymbol{\rm μ≠0}$时，该算符具有唯一的单离散本征值$\boldsymbol{\rm E_γ(k,μ)=2(1+γ)+\operatorname{sgn}(μ)\sqrt{μ^2+4a_γ(k)^2}}$：当$\boldsymbol{\rm μ<0}$时位于能带下方，$\boldsymbol{\rm μ>0}$时位于能带上方，对应的本征函数呈指数衰减。在例外纤维$\boldsymbol{\rm (γ,k)=(1,π)}$处，能带坍缩为$\boldsymbol{\rm \{4\}}$，唯一的单本征值为$\boldsymbol{\rm E_1(π,μ)=4+μ}$。\n在临界耦合$\boldsymbol{\rm μ=0}$且$\boldsymbol{\rm a_γ(k)>0}$时，两个谱边均为阈值共振，对应有界的非平方可积解。当$\boldsymbol{\rm μ→0^±}$时，离散本征值趋近于相应阈值，满足$\boldsymbol{\rm |E_γ(k,μ)-E_{\mathrm{thr}}^±(k)| =\n\frac{μ^2}{4a_γ(k)}+O(μ^4)}$，且归一化本征函数收敛于共振解。强耦合下，$\boldsymbol{\rm E_γ(k,μ)=μ+2(1+γ)+O(|μ|^{-1})}$，本征函数局域在相互作用格点处。我们还根据$\boldsymbol{\rm |μ|}$与$\boldsymbol{\rm a_γ(k)}$的相对标度，对联合极限$\boldsymbol{\rm μ→0^±}$、$\boldsymbol{\rm (γ,k)→(1,π)}$进行了分类，揭示了弱耦合阈值渐近行为的非一致性。

英文摘要

We study a two-particle lattice Schrödinger operator describing a fermion-boson pair on the one-dimensional lattice $\mathbb Z$ with zero-range on-site interaction of strength $μ\in\mathbb R$ and mass ratio $γ>0$. Using relative coordinates and fiber decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-π,π]$, we obtain a real symmetric fiber Hamiltonian $H_{μ,γ}(k)$ in $\ell^2(\mathbb Z)$ with essential spectrum $$ [\,2(1+γ)-2a_γ(k),\,2(1+γ)+2a_γ(k)\,], \qquad a_γ(k)=\sqrt{1+2γ\cos k+γ^2}. $$ For $a_γ(k)>0$ and $μ\ne0$, the operator has a unique simple discrete eigenvalue $$ E_γ(k,μ)=2(1+γ)+\operatorname{sgn}(μ)\sqrt{μ^2+4a_γ(k)^2}, $$ lying below the band for $μ<0$ and above it for $μ>0$, with an exponentially decaying eigenfunction. At the exceptional fiber $(γ,k)=(1,π)$, the band collapses to ${4}$ and the unique simple eigenvalue is $E_1(π,μ)=4+μ$. At the critical coupling $μ=0$ and $a_γ(k)>0$, both spectral edges are threshold resonances with bounded non-square-integrable solutions. As $μ\to0^\pm$, the discrete eigenvalue approaches the corresponding threshold with $$ |E_γ(k,μ)-E_{\mathrm{thr}}^\pm(k)| =\frac{μ^2}{4a_γ(k)}+O(μ^4), $$ and the normalized eigenfunction converges to the resonant solution. For strong coupling, $E_γ(k,μ)=μ+2(1+γ)+O(|μ|^{-1})$, while the eigenfunction localizes at the interaction site. We also classify the joint limit $μ\to0^\pm$, $(γ,k)\to(1,π)$ according to the relative scale of $|μ|$ and $a_γ(k)$, revealing the nonuniformity of the weak-coupling threshold asymptotics.

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18 pages