Boundary $c\leq1$ Liouville CFT from the Bootstrap

We construct boundary Liouville conformal field theory at central charge $c\leq1$ with FZZT-type boundary conditions and show that it satisfies the Cardy--Lewellen sewing constraints on the disk and the annulus. Using the analytic bootstrap we derive the so far missing boundary three-point and bulk-to-boundary structure constants and a consistent disk one-point function as explicit meromorphic functions of the Liouville momenta. In sharp contrast to $c\geq25$, the open-string spectrum between two ``timelike'' FZZT boundaries is \emph{discrete}, yet non-trivially consistent through open-closed duality with the \emph{continuous} closed-string spectrum of Ribault and Santachiara.