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Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

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Abstract

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the operators h(k) depends on the particle interaction energy and the total quasimomentum \(k \in \mathbb{T}^3\) (where \(\mathbb{T}^3\) is a three-dimensional torus). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator h(0).

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Correspondence to M. E. Muminov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 180, No. 3, pp. 329–341, July, 2014.

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Muminov, M.E., Khurramov, A.M. Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice. Theor Math Phys 180, 1040–1050 (2014). https://doi.org/10.1007/s11232-014-0198-2

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Keywords

  • two-particle Hamiltonian on a lattice
  • virtual level
  • virtual level multiplicity
  • eigenvalue