Abstract
For a wide class of two-particle Schrödinger operators H(k) = H0(k) + V, k ∈ \(\mathbb{T}^d\), corresponding to a two-fermion system on a d-dimensional cubic integer lattice (d ≥ 1), we prove that for any value k ∈ \(\mathbb{T}^d\) 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.
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This research was supported by the Foundation for Basic Research of the Republic of Uzbekistan (Grant No. OTF4-66).
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 2, pp. 251–268, May, 2020.
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Lakaev, S.N., Abdukhakimov, S.H. Threshold effects in a two-fermion system on an optical lattice. Theor Math Phys 203, 648–663 (2020). https://doi.org/10.1134/S0040577920050074
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DOI: https://doi.org/10.1134/S0040577920050074