We consider the self-adjoint Smilansky Hamiltonian Hξ in L2(R2) associated with the formal differential expression -∂2x – ½(∂2y +y2)- √2 ξyδ(x) in the sub-critical regime, ξ ϵ(0, 1). We demonstrate the existence of resonances for Hξ 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 characterize resonances for small ξ>0. In addition, we refine the previously known results on the bound states of Hξ in the weak coupling regime (ξ→ 0+). In the proofs we use Birman-Schwinger principle for Hξ ,elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.