Spectrum of the~two-particle Schr\"odinger operator on a~lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 2, pp. 287-300
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We consider the family of two-particle discrete Schrödinger operators
$H(k)$ associated with the Hamiltonian of a system of two fermions on
a $\nu$-dimensional lattice $\mathbb Z^{\nu}$, $\nu\geq 1$, where
$k\in\mathbb T^{\nu}\equiv(-\pi,\pi]^{\nu}$ is a two-particle quasimomentum. We
prove that the operator $H(k)$, $k\in\mathbb T^{\nu}$, $k\ne0$, has an eigenvalue
to the left of the essential spectrum for any dimension $\nu=1,2,\dots$ if
the operator $H(0)$ has a virtual level ($\nu=1,2$) or an eigenvalue
($\nu\geq 3$) at the bottom of the essential spectrum (of the two-particle
continuum).
Keywords:
spectral properties, two-particle discrete Schrödinger operator, Birman–Schwinger principle, virtual level, eigenvalue.
@article{TMF_2008_155_2_a7,
author = {S. N. Lakaev and A. M. Khalkhuzhaev},
title = {Spectrum of the~two-particle {Schr\"odinger} operator on a~lattice},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {287--300},
publisher = {mathdoc},
volume = {155},
number = {2},
year = {2008},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2008_155_2_a7/}
}
TY - JOUR AU - S. N. Lakaev AU - A. M. Khalkhuzhaev TI - Spectrum of the~two-particle Schr\"odinger operator on a~lattice JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2008 SP - 287 EP - 300 VL - 155 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2008_155_2_a7/ LA - ru ID - TMF_2008_155_2_a7 ER -
S. N. Lakaev; A. M. Khalkhuzhaev. Spectrum of the~two-particle Schr\"odinger operator on a~lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 2, pp. 287-300. http://geodesic.mathdoc.fr/item/TMF_2008_155_2_a7/