  We consider the self-adjoint Smilansky Hamiltonian $\mathsf{H}_\varepsilon$
in $L^2(\mathbb{R}^2)$ associated with the formal differential expression
$-\partial_x^2 - \frac12\big(\partial_y^2 + y^2) - \sqrt{2}\varepsilon y
\delta(x)$ in the sub-critical regime, $\varepsilon \in (0,1)$. We demonstrate
the existence of resonances for $\mathsf{H}_\varepsilon$ on a countable
subfamily of sheets of the underlying Riemann surface whose distance from the
physical sheet is finite. On such sheets, we find resonance free regions and
characterise resonances for small $\varepsilon > 0$. In addition, we refine the
previously known results on the bound states of $\mathsf{H}_\varepsilon$ in the
weak coupling regime ($\varepsilon\rightarrow 0+$). In the proofs we use
Birman-Schwinger principle for $\mathsf{H}_\varepsilon$, elements of spectral
theory for Jacobi matrices, and the analytic implicit function theorem.
