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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (5), P. 789–802

On resonances and bound states of Smilansky Hamiltonian

P. Exner – Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic; exner@ujf.cas.cz
V. Lotoreichik – Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic; lotoreichik@ujf.cas.cz
M. Tater – Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic; tater@ujf.cas.cz

We consider the self-adjoint Smilansky Hamiltonian H_ε in L² R²) associated with the formal differential expression -∂_x²-1/2(∂_y²+y²)-√2εyδ(x) in the sub-critical regime, ε∈(0, 1). We demonstrate the existence of resonances for H_ε 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 ε > 0. In addition, we refine the previously known results on the bound states of H_ε in the weak coupling regime (ε→0+). In the proofs we use Birman-Schwinger principle for H_ε, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

Keywords: Smilansky Hamiltonian, resonances, resonance free region, weak coupling asymptotics, Riemann surface, bound states.

PACS 02.30.Tb, 03.65.Db

DOI 10.17586/2220-8054-2016-7-5-789-802

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