  Exact solvability (ES) of one-dimensional quantum potentials $V(x)$ is a
vague concept. We propose that beyond its most conventional range the ES status
should be attributed also to many less common interaction models for which the
wave functions remain piecewise proportional to special functions. The claim is
supported by constructive analysis of a toy model $V(x)= -g^2\exp (-|x|)$. The
detailed description of the related bound-state and scattering solutions of
Schr\"{o}dinger equation is provided in terms of Bessel functions which are
properly matched in the origin.
