On resonances and bound states of Smilansky Hamiltonian

Abstract

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.