Geodesic
Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 170 (2012) no. 3, pp. 393-408 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the two-particle discrete Schrödinger operator $H_\mu(K)$ corresponding to a system of two arbitrary particles on a $d$-dimensional lattice $\mathbb Z^d$, $d\ge3$, interacting via a pair contact repulsive potential with a coupling constant $\mu>0$ ($K\in\mathbb T^d$ is the quasimomentum of two particles). We find that the upper (right) edge of the essential spectrum can be either a virtual level (for $d=3,4)$ or an eigenvalue (for $d\ge5)$ of $H_\mu(K)$. We show that there exists a unique eigenvalue located to the right of the essential spectrum, depending on the coupling constant $\mu$ and the two-particle quasimomentum $K$. We prove the analyticity of the corresponding eigenstate and the analyticity of the eigenvalue and the eigenstate as functions of the quasimomentum $K\in\mathbb T^d$ in the domain of their existence.
Keywords: discrete Schrödinger operator, two-particle system, Hamiltonian, contact repulsive potential, virtual level, eigenvalue, lattice. 
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     author = {S. N. Lakaev and S. S. Ulashov},
     title = {Existence and analyticity of bound states of a~two-particle {Schr\"odinger} operator on a~lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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S. N. Lakaev; S. S. Ulashov. Existence and analyticity of bound states of a two-particle Schrödinger operator on a lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 170 (2012) no. 3, pp. 393-408. http://geodesic.mathdoc.fr/item/TMF_2012_170_3_a5/

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