一维晶格上两个全同费米子的显式束缚态与阈值共振
Explicit Bound States and Threshold Resonances for Two Identical Fermions on a One-Dimensional Lattice
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中文总结 AI 辅助
本研究针对一维晶格上带最近邻相互作用的两全同费米子系统，通过直积分分解约化问题，完整刻画了谱边跃迁、束缚态存在条件及阈值共振特性，并推导了渐近行为与例外纤维情况。
中文摘要 AI 辅助
我们研究了描述一维晶格$\u2124$上两个全同费米子的两粒子晶格薛定谔算符，其最近邻相互作用强度为$\u03bb\u2208\u211d$。利用关于总准动量$k\u2208\u211c:=(-\u03c0,\u03c0]$的直积分分解，我们将问题约化为作用于奇相对坐标空间的纤维算符$H_\u03bb(k)$。对于$k\u2208(-\u03c0,\u03c0)$，其本质谱为$\u03c3_{\u2113\u211f\u211f}(H_\u03bb(k)) = [4-4\uc0ac(k/2), 4+4\uc0ac(k/2)]$。我们完整描述了两个谱边处的谱跃迁。当且仅当$|\u03bb|>2\uc0ac(k/2)$时，该算符在本质谱外存在唯一的单本征值，此时本征值为$E(k,\u03bb) = 4+\u03bb+\frac{4\uc0ac^2(k/2)}{\u03bb}$。当$|\u03bb|<2\uc0ac(k/2)$时，不存在离散本征值。在临界耦合$|\u03bb|=2\uc0ac(k/2)$处，本征值与对应的谱边合并，成为阈值共振，对应有界的非平方可积奇解。我们还推导了本征值与本征函数的阈值渐近行为和强耦合渐近行为。在强耦合区域，本征函数局域于相互作用格点处；而在临界耦合处，它逐点收敛于对应的共振解。我们单独处理了例外纤维$k=\u03c0$的情况，此时跃迁项消失，本质谱坍缩为$\u20224\u2022$。
英文摘要
We study a two-particle lattice Schrödinger operator describing two identical fermions on the one-dimensional lattice $\mathbb Z$ with nearest-neighbor interaction of strength $λ\in\mathbb R$. Using the direct-integral decomposition with respect to the total quasi-momentum $k\in\mathbb T:=(-π,π]$, we reduce the problem to fiber operators $H_λ(k)$ acting in the odd relative-coordinate space. For $k\in(-π,π)$, the essential spectrum is \[ σ_{\mathrm{ess}}(H_λ(k)) = [4-4\cos(k/2),\,4+4\cos(k/2)]. \] We give a complete description of the spectral transition at both edges. The operator has a unique simple eigenvalue outside the essential spectrum if and only if \[ |λ|>2\cos(k/2), \] in which case \[ E(k,λ) = 4+λ+\frac{4\cos^2(k/2)}λ. \] For $|λ|<2\cos(k/2)$ there is no discrete eigenvalue. At the critical coupling $|λ|=2\cos(k/2)$, the eigenvalue merges with the corresponding spectral edge and becomes a threshold resonance, with a bounded non-square-integrable odd solution. We also derive the threshold and strong-coupling asymptotics of the eigenvalue and eigenfunction. In the strong-coupling regime the eigenfunction localizes at the interaction sites, whereas at critical coupling it converges pointwise to the corresponding resonant solution. The exceptional fiber $k=π$, where the hopping vanishes and the essential spectrum collapses to $\{4\}$, is treated separately.
