  We study the P\"oschl-Teller potential $V(x) = \alpha^2 g_s \sinh^{-2}(\alpha
x) + \alpha^2 g_c \cosh^{-2}(\alpha x)$, for every value of the dimensionless
parameters $g_s$ and $g_c$, including the less usual ranges for which the
regular singularity at the origin prevents the Hamiltonian from being
self-adjoint. We apply a renormalization procedure to obtain a family of
well-defined energy eigenfunctions, and study the associated renormalization
group (RG) flow. We find an anomalous length scale that appears by dimensional
transmutation, and spontaneously breaks the asymptotic conformal symmetry near
the singularity, which is also explicitly broken by the dimensionful parameter
$\alpha$ in the potential. These two competing ways of breaking conformal
symmetry give the RG flow a rich structure, with phenomena such as a possible
region of walking coupling, massive phases, and non-trivial limits even when
the anomalous dimension is absent. We show that supersymmetry of the potential,
when present, is also spontaneously broken, along with asymptotic conformal
symmetry. We use the family of eigenfunctions to compute the S-matrix in all
regions of parameter space, for any value of anomalous scale, and
systematically study the poles of the S-matrix to classify all bound,
anti-bound and metastable states, including quasi-normal modes. The anomalous
scale, as expected, changes the spectra in non-trivial ways.
