Quantum JT Gravity in a box as a Pöschl-Teller Scattering Problem

We present a canonical quantization of Jackiw-Teitelboim gravity with finite Dirichlet boundary conditions, using the geodesic length between the two boundaries and its conjugate momentum as reduced phase space variables. The dynamics is recast as the scattering problem of a nonrelativistic particle in a repulsive P\"oschl-Teller potential, naturally embedded within a hyperbolic reduction of the $\mathfrak{sl}(2,\mathbb{R})$ Casimir. We obtain exact wavefunctions of the universe and the disk partition function, interpreted as a transition matrix element between states of vanishing bare length. In the asymptotic limit, the theory reduces to Liouville quantum mechanics and reproduces the standard Schwarzian spectral density. At finite cutoff, however, the spectral measure exhibits genuinely nonperturbative corrections, absent in existing $T\bar T$ treatments. We also obtain closed form expressions for thermal two-point functions in terms of Wilson functions and propose diagrammatic rules for time- and out-of-time-ordered four-point functions. We further address the issue of the branch cut singularity of the quasi-local energy and propose a UV completion of the model in which the Brown-York charge is analytically continued beyond the black hole horizon. This continuation naturally extends the scattering problem to configurations that foliate the black hole interior.