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We investigate bound states in the one-dimensional two-particle Bose-Hubbardmodel with an attractive ($V> 0$) impurity potential. This is aone-dimensional, discrete analogy of the hydrogen negative ion H$^-$ problem.There are several different types of bound states in this system, each of whichappears in a specific region. For given $V$, there exists a (positive) criticalvalue $U_{c1}$ of $U$, below which the ground state is a bound state.Interestingly, close to the critical value ($U\lesssim U_{c1}$), the groundstate can be described by the Chandrasekhar-type variational wave function,which was initially proposed for H$^-$. For $U>U_{c1}$, the ground state is nolonger a bound state. However, there exists a second (larger) critical value$U_{c2}$ of $U$, above which a molecule-type bound state is established andstabilized by the repulsion. We have also tried to solve for the eigenstates ofthe 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 ofthe states in the even-parity sector. This allows us to identify analyticallytwo 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 ofthese regions, there exists an even-parity bound state accompanying thecorresponding odd-parity bound state with almost the same energy.

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