  The regularized trace of the heat kernel of a one-dimensional Schr\"odinger
operator with a singular two-particle contact interaction being of Lieb-Liniger
type is considered. We derive a complete small-time asymptotic expansion in
(fractional) powers of the time, $t$. Most importantly, we do not invoke
standard parametrix constructions for the heat kernel. Instead, we first derive
the large-energy expansion of the regularized trace of the resolvent for the
considered operator. Then, we exploit that the resolvent may be obtained by a
Laplace transformation of the heat semi-group, and an application of a suitable
inverse Watson lemma eventually yields the small-$t$ asymptotic expansion of
the heat-kernel trace.
