Geodesic
Spectrum of the two-particle Schrödinger 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. 
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     title = {Spectrum of the~two-particle {Schr\"odinger} operator on a~lattice},
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S. N. Lakaev; A. M. Khalkhuzhaev. Spectrum of the two-particle Schrödinger 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/

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