On resonances and bound states of Smilansky Hamiltonian

We consider the self-adjoint Smilansky Hamiltonian H$_\varepsilon$ in L$^2(\mathbb{R}^2)$ associated with the formal differential expression $-\partial^2_x-1/2(\partial^2_y+y^2)-\sqrt2\varepsilon y\delta(x)$ in the sub-critical regime, $\varepsilon\in(0,1)$. We demonstrate the existence of resonances for 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 characterize resonances for small $\varepsilon>0$. In addition, we refine the previously known results on the bound states of H$_\varepsilon$, in the weak coupling regime $(\varepsilon\to0+)$. In the proofs we use Birman–Schwinger principle for H$_\varepsilon$, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.