Geodesic
On resonances and bound states of Smilansky Hamiltonian
Nanosistemy: fizika, himiâ, matematika, Tome 7 (2016) no. 5, pp. 789-802.

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We consider the self-adjoint Smilansky Hamiltonian H$_\varepsilon$ in L$^2(\mathbb{R}^2)$ associated with the formal differential expression $-\partial^2_x-1/2(\partial^2_y+y^2)-\sqrt2\varepsilon y\delta(x)$ in the sub-critical regime, $\varepsilon\in(0,1)$. We demonstrate the existence of resonances for 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 characterize resonances for small $\varepsilon>0$. In addition, we refine the previously known results on the bound states of H$_\varepsilon$, in the weak coupling regime $(\varepsilon\to0+)$. In the proofs we use Birman–Schwinger principle for H$_\varepsilon$, 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. 
@article{NANO_2016_7_5_a1,
     author = {P. Exner and V. Lotoreichik and M. Tater},
     title = {On resonances and bound states of {Smilansky} {Hamiltonian}},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
     pages = {789--802},
     publisher = {mathdoc},
     volume = {7},
     number = {5},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2016_7_5_a1/}
}
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P. Exner; V. Lotoreichik; M. Tater. On resonances and bound states of Smilansky Hamiltonian. Nanosistemy: fizika, himiâ, matematika, Tome 7 (2016) no. 5, pp. 789-802. http://geodesic.mathdoc.fr/item/NANO_2016_7_5_a1/