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We revisit the Lieb-Liniger model for $n$ bosons in one dimension withattractive delta interaction in a half-space $\mathbb{R}^+$ with diagonalboundary conditions. This model is integrable for arbitrary value of $b \in\mathbb{R}$, the interaction parameter with the boundary. We show that itsspectrum exhibits a sequence of transitions, as $b$ is decreased from thehard-wall case $b=+\infty$, with successive appearance of boundary bound states(or boundary modes) which we fully characterize. We apply these results tostudy the Kardar-Parisi-Zhang equation for the growth of a one-dimensionalinterface of height $h(x,t)$, on the half-space with boundary condition$\partial_x h(x,t)|_{x=0}=b$ and droplet initial condition at the wall. Weobtain explicit expressions, valid at all time $t$ and arbitrary $b$, for theinteger exponential (one-point) moments of the KPZ height field $\bar{e^{nh(0,t)}}$. From these moments we extract the large time limit of theprobability distribution function (PDF) of the scaled KPZ height function. Itexhibits a phase transition, related to the unbinding to the wall of theequivalent directed polymer problem, with two phases: (i) unbound for$b>-\frac{1}{2}$ where the PDF is given by the GSE Tracy-Widom distribution(ii) bound for $b<-\frac{1}{2}$, where the PDF is a Gaussian. At the criticalpoint $b=-\frac{1}{2}$, the PDF is given by the GOE Tracy-Widom distribution.

Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions