On resonances and bound states of Smilansky Hamiltonian

TL;DR

This paper investigates the spectral properties of the Smilansky Hamiltonian, revealing the existence and characterization of resonances and refining the understanding of bound states in the weak coupling limit.

Contribution

It demonstrates the existence of resonances on specific Riemann surface sheets and refines the analysis of bound states for small coupling parameters.

Findings

Resonances exist on countable sheets of the Riemann surface.

Resonance free regions are identified for small coupling.

Bound states are characterized in the weak coupling regime.

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.