  The Efimov effect (in a broad sense) refers to the onset of a geometric
sequence of many-body bound states as a consequence of the breakdown of
continuous scale invariance to discrete scale invariance. While originally
discovered in three-body problems in three dimensions, the Efimov effect has
now been known to appear in a wide spectrum of many-body problems in various
dimensions. Here we introduce a simple, exactly solvable toy model of two
identical bosons in one dimension that exhibits the Efimov effect. We consider
the situation where the bosons reside on a semi-infinite line and interact with
each other through a pairwise $\delta$-function potential with a particular
position-dependent coupling strength that makes the system scale invariant. We
show that, for sufficiently attractive interaction, the bosons are bound
together and a new energy scale emerges. This energy scale breaks continuous
scale invariance to discrete scale invariance and leads to the onset of a
geometric sequence of two-body bound states. We also study the two-body
scattering off the boundary and derive the exact reflection amplitude that
exhibits a log-periodicity. This article is intended for students and
non-specialists interested in discrete scale invariance.
