On resonances and bound states of Smilansky Hamiltonian

0466593 - ÚJF 2017 RIV RU eng J - Journal Article
Exner, Pavel - Lotoreichik, Vladimir - Tater, Miloš
On resonances and bound states of Smilansky Hamiltonian.
Nanosystems: Physics, Chemistry, Mathematics. Roč. 7, č. 5 (2016), s. 789-802. ISSN 2220-8054
R&D Projects: GA ČR(CZ) GA14-06818S
Institutional support: RVO:61389005
Keywords : Smilansky Hamiltonian * resonances * resonance free region * weak coupling asymptotics * Riemann surface * bound states
Subject RIV: BE - Theoretical Physics

We consider the self-adjoint Smilansky Hamiltonian H epsilon in L-2(R-2) associated with the formal differential expression -partial derivative(2)(x) - 1/2 (partial derivative(2)(y) + y(2)) - root 2 epsilon y delta(x) in the sub-critical regime, epsilon is an element of (0, 1). We demonstrate the existence of resonances for H-epsilon 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 epsilon > 0. In addition, we refine the previously known results on the bound states of H " in the weak coupling regime (epsilon -> 0+). In the proofs we use Birman-Schwinger principle for H-epsilon, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.
Permanent Link: http://hdl.handle.net/11104/0264857