16-44 Pavel Exner, Vladimir Lotoreichik, Milos Tater

On resonances and bound states of Smilansky Hamiltonian (285K, pdf) Jul 2, 16

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Abstract. We consider the self-adjoint Smilansky Hamiltonian $\Op_\eps$ in $L^2(\dR^2)$ associated with the formal differential expression $-\p_x^2 - rac12ig(\p_y^2 + y^2) - \sqrt{2}\eps y \delta(x)$ in the sub-critical regime, $\eps \in (0,1)$. We demonstrate the existence of resonances for $\Op_\eps$ 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 $\eps > 0$. In addition, we refine the previously known results on the bound states of $\Op_\eps$ in the weak coupling regime ($\epsrr 0+$). In the proofs we use Birman-Schwinger principle for $\Op_\eps$, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

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