On resonances and bound states of Smilansky Hamiltonian

Abstract

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