Эта публикация цитируется в
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On resonances and bound states of Smilansky Hamiltonian
P. Exner,
V. Lotoreichik,
M. Tater Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech Republic
Аннотация:
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.
Ключевые слова:
Smilansky Hamiltonian, resonances, resonance free region, weak coupling asymptotics, Riemann surface, bound states.
PACS:
02.30.Tb,
03.65.Db Поступила в редакцию: 01.07.2016
Исправленный вариант: 28.07.2016
Язык публикации: английский
DOI:
10.17586/2220-8054-2016-7-5-789-802