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; email@example.com
V. Lotoreichik – Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic; firstname.lastname@example.org
M. Tater – Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic; email@example.com
We consider the self-adjoint Smilansky Hamiltonian Hε 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 Hε 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 Hε in the weak coupling regime (ε→0+). In the proofs we use Birman-Schwinger principle for Hε, 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