 Solution

For $E<0$, inversion of the free operator gives $G_0(x,y)=m{e}^{-\kappa |x-y|}/(\kappa {\hslash }^2)$, with $\kappa =\sqrt{-2mE}/\hslash$. The bound-state equation is $\psi =-G_{0}U\psi$, henceFor the separable kernel $V(y,z)=-\alpha F(y)F^{\ast }(z)$, defineThenAn overall normalization still has to be chosen after finding a permitted energy. A nontrivial bound state must have $C\ne 0$: if $C=0$, the potential acts as zero and a square-integrable negative-energy free solution on the whole line cannot exist. Decay or square-integrability assumptions on $F$ are needed to make these integrals well defined; smoothness alone would not ensure convergence.