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Bound states in the one-dimensional two-particle Hubbard model with an impurity

We investigate bound states in the one-dimensional two-particle Bose-Hubbard model with an attractive ($V> 0$) impurity potential. This is a one-dimensional, discrete analogy of the hydrogen negative ion H$^-$ problem. There are several different types of bound states in this system, each of which appears in a specific region. For given $V$, there exists a (positive) critical value $U_{c1}$ of $U$, below which the ground state is a bound state. Interestingly, close to the critical value ($U\lesssim U_{c1}$), the ground state can be described by the Chandrasekhar-type variational wave function, which was initially proposed for H$^-$. For $U>U_{c1}$, the ground state is no longer a bound state. However, there exists a second (larger) critical value $U_{c2}$ of $U$, above which a molecule-type bound state is established and stabilized by the repulsion. We have also tried to solve for the eigenstates of the model using the Bethe ansatz. The model possesses a global $\Zz_2$-symmetry (parity) which allows classification of all eigenstates into even and odd ones. It is found that all states with odd-parity have the Bethe form, but none of the states in the even-parity sector. This allows us to identify analytically two odd-parity bound states, which appear in the parameter regions $-2V<U<-V$ and $-V<U<0$, respectively. Remarkably, the latter one can be \textit{embedded} in the continuum spectrum with appropriate parameters. Moreover, in part of these regions, there exists an even-parity bound state accompanying the corresponding odd-parity bound state with almost the same energy.