Lyapunov Exponents of the Half-Line SHE

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer) Lyapunov exponents of SHE. As a consequence, prove large deviation principle for upper tail KPZ under Neumann -\frac{1}{2}$ rate function $\Phi_+^{\text{hf}} (s) \frac{2}{3} s^{\frac{3}{2}}$. This confirms prediction [Krajenbrink and Le Doussal 2018] [Meerson, Vilenkin exponent equation.