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The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice

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Abstract

We consider the Hamiltonian \( \hat h_{\mu \lambda } ,\mu ,\lambda \geqslant 0 \), describing the motion of one quantum particle on a three-dimensional lattice in an external field. We investigate the number of eigenvalues and their arrangement depending on the value of the interaction energy for µ ≥ 0 and λ ≥ 0.

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Correspondence to S. N. Lakaev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 425–443, March, 2009.

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Lakaev, S.N., Bozorov, I.N. The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice. Theor Math Phys 158, 360–376 (2009). https://doi.org/10.1007/s11232-009-0030-6

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Keywords

  • one-particle Hamiltonian
  • continuous spectrum
  • virtual level
  • eigenvalue
  • Birman-Schwinger operator
  • Fredholm determinant