Geodesic
Threshold effects in a~two-fermion system on an~optical lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 2, pp. 251-268

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For a wide class of two-particle Schrödinger operators $H(k)=H_0(k)+V$, $k\in\mathbb T^d$, corresponding to a two-fermion system on a $d$-dimensional cubic integer lattice $(d\ge1)$, we prove that for any value $k\in\mathbb T^d$ of the quasimomentum, the discrete spectrum of $H(k)$ below the lower threshold of the essential spectrum is a nonempty set if the following two conditions are satisfied. First, the two-particle operator $H(0)$ corresponding to a zero quasimomentum has either an eigenvalue or a virtual level on the lower threshold of the essential spectrum. Second, the one-particle free (nonperturbed) Schrödinger operator in the coordinate representation generates a semigroup that preserves positivity.
Keywords: two-fermion system, discrete Schrödinger operator, Hamiltonian, conditionally negative-definite function, virtual level, bound state.
Mots-clés : dispersion relation 
@article{TMF_2020_203_2_a6,
     author = {S. N. Lakaev and S. Kh. Abdukhakimov},
     title = {Threshold effects in a~two-fermion system on an~optical lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {251--268},
     publisher = {mathdoc},
     volume = {203},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_203_2_a6/}
}
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S. N. Lakaev; S. Kh. Abdukhakimov. Threshold effects in a~two-fermion system on an~optical lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 2, pp. 251-268. http://geodesic.mathdoc.fr/item/TMF_2020_203_2_a6/