  We show that for a particular model, the quantum mechanical bootstrap is
capable of finding exact results. We consider a solvable system with
Hamiltonian $H=SZ(1-Z)S$, where $Z$ and $S$ satisfy canonical commutation
relations. While this model may appear unusual, using an appropriate coordinate
transformation, the Schr\"odinger equation can be cast into a standard form
with a P\"oschl-Teller-type potential. Since the system is defined on an
interval, it is well-known that $S$ is not self-adjoint. Nevertheless, the
bootstrap method can still be implemented, producing an infinite set of
positivity constraints. Using a certain operator ordering, the energy
eigenvalues are only constrained into bands. With an alternative ordering,
however, we find that a finite number of constraints is sufficient to fix the
low-lying energy levels exactly.
