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Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

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Abstract

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice 3 (here k is the total quasimomentum of a two-particle system, \(k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)\). We show that for any \(k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3\), there is a potential \(\hat v\) such that the two-particle operator H(k) has infinitely many eigenvalues zn(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ⊂ S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k 2 , k 3 ) ∈ (−π,π)2, and \(\hat v \in W(3)\), then the operator H(k) has infinitely many eigenvalues zn(k) with an asymptotic exponential form as n → and if k ∈ S(i) ∩ S(j) and \(\hat v \in W(ij)\), then the eigenvalues znm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.

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Correspondence to J. I. Abdullaev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 3, pp. 417–428, September 2013.

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Abdullaev, J.I., Mamirov, B.U. Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator. Theor Math Phys 176, 1184–1193 (2013). https://doi.org/10.1007/s11232-013-0099-9

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Keywords

  • Hamiltonian
  • total quasimomentum
  • Schrödinger operator
  • asymptotic behavior
  • eigenvalue
  • eigenfunction