  We first quantize the action proposed by Casalbuoni and Gomis in [Phys. Rev.
D \textbf{90}, 026001 (2014)], an action that describes two massless
relativistic scalar particles interacting via a conformally invariant
potential. The spectrum is a continuum of massive states that may be
interpreted as unparticles. We then obtain in a similar way the mass operator
for a deformed action in which two terms are introduced that break the
conformal symmetry: a mass term and an extra position-dependent coupling
constant. A simple Ansatz for the latter leads to a mass operator with linear
confinement in terms of an effective string tension $\sigma\,$. The quantized
model is confining when $\sigma\neq0$ and its mass spectrum shows Regge
trajectories. We propose a tensionless limit in which highly excited confined
states reduce to (gapped) unparticles. Moreover, the low-lying confined bound
states become massless in the latter limit as a sign of conformal symmetry
restoration and the ratio between their masses and $\sqrt\sigma$ stays
constant. The originality of our approach is that it applies to both confining
and conformal phases via an effective interacting model.
