We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value of , the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case , with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar–Parisi–Zhang equation for the growth of a one-dimensional interface of height , on the half-space with boundary condition and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b, for the integer exponential (one-point) moments of the KPZ height field . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for where the PDF is given by the GSE Tracy–Widom distribution (ii) bound for , where the PDF is a Gaussian. At the critical point , the PDF is given by the GOE Tracy–Widom distribution.

中文翻译：

---

半线上的 Delta-Bose 气体和 Kardar-Parisi-Zhang 方程：边界束缚态和非束缚跃迁

我们重新审视了具有对角边界条件的半空间中具有吸引力的 delta 相互作用的一维 n 玻色子的 Lieb-Liniger 模型。该模型对于 的任意值是可积的，即与边界的交互参数。我们表明，随着 b 从硬壁情况 减小，其光谱表现出一系列跃迁，连续出现边界束缚态（或边界模式），我们完全表征了这些。我们应用这些结果来研究 Kardar-Parisi-Zhang 方程，用于在具有边界条件和壁处液滴初始条件的半空间上，高度为 的一维界面的增长。对于 KPZ 高度场 的整数指数（一点）矩，我们获得了在所有时间 t 和任意时间 b 都有效的显式表达式。从这些时刻，我们提取了缩放 KPZ 高度函数的概率分布函数 (PDF) 的大时间限制。它表现出相变，与等效定向聚合物问题的壁解绑定相关，具有两个阶段：(i) 不受约束，其中 PDF 由 GSE Tracy-Widom 分布给出 (ii) 约束 ，其中 PDF是一个高斯。在临界点 ，PDF 由 GOE Tracy-Widom 分布给出。