Threshold spectral transition for a fermion-boson pair on the one-dimensional lattice

Abstract

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