DISCRETE SPECTRUM OF TWO-FERMION SCHRÖDINGER OPERATORS ON 3D LATTICES

Authors

  • Saidakbar Abdukhakimov Samarkand Branch of the Romanovskii Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, PhD Author
  • A.B. Khasanov Samarkand Branch of the Romanovskii Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, PhD Author

Keywords:

Lattice Schrödinger, fundamental, physics

Abstract

This thesis investigates the spectral properties of two-fermion Schrödinger operators on the three-dimensional lattice  with nearest- and next-nearest-neighbor interactions. We focus on the discrete eigenvalues lying outside the essential spectrum and analyze their dependence on interaction parameters. The study employs decomposition into invariant subspaces and Fredholm determinant techniques to characterize bound states and critical phenomena.

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Author Biography

  • A.B. Khasanov, Samarkand Branch of the Romanovskii Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, PhD
    • Albeverio S., Lakaev S. N., Makarov K. A., Muminov Z. I. The Threshold Effects for the Two-particle Hamiltonians on Lattices. Communications in Mathematical Physics, 2006, 262, P. 91–115.
    • Albeverio S., Lakaev S. N., Makarov K. A., Muminov Z. I. The Threshold Effects for the Two-particle Hamiltonians on Lattices. Communications in Mathematical Physics, 2006, 262, P. 91–115.
    • Bach V., de Siqueira Pedra W., Lakaev S.N. Bounds on the discrete spectrum of lattice Schro¨dinger operators. Journal of Mathematical Physics, (2017), 59(2), 022109.
    • Faria Da Veiga P.A., Ioriatti L., O’Carroll M. Energy-momentum spectrum of some two-particle lattice Schro¨dinger Hamiltonians. Physical Review E, (2002), 66 , 016130.
    • Kholmatov Sh.Yu., Lakaev S. N., Almuratov F. Bound states of discrete Schro¨dinger operators on one and two dimensional lattices. Math. Anal. Appl.,(2021), 503(1), 125280.
    • Lakaev S.N., Abdukhakimov S.Kh. Threshold effects in a two-fermion system on an optical lattice. and Math. Phys., (2020), 203(2), P. 251–268
    • Lakaev S.N., Kholmatov Sh.Yu., Khamidov Sh.I. Bose-Hubbard model with onsite and nearest-neighbor interactions; exactly solvable case. Phys. A: Math. Theor.,(2021), 54, 245201.
    • Lakaev S. N., Dell’Antonio G., Khalkhuzhaev A. Existence of an isolated band in a system of three particles in an optical lattice. Phys. A: Math. Theor., (2016), 49(52) P. 15.
    • Lakaev S.N., Lakaev Sh.S. The existence of bound states in a system of three particles in an optical lattice. Phys. A: Math. Theor., (2017), 50, 335202.
    • Lakaev S.N., Motovilov A.K., Abdukhakimov S.Kh. Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions. Phys. A: Math., (2023), 56, 315202.

References

Albeverio S., Lakaev S. N., Makarov K. A., Muminov Z. I. The Threshold Effects for the Two-particle Hamiltonians on Lattices. Communications in Mathematical Physics, 2006, 262, P. 91–115.

Albeverio S., Lakaev S. N., Makarov K. A., Muminov Z. I. The Threshold Effects for the Two-particle Hamiltonians on Lattices. Communications in Mathematical Physics, 2006, 262, P. 91–115.

Bach V., de Siqueira Pedra W., Lakaev S.N. Bounds on the discrete spectrum of lattice Schro¨dinger operators. Journal of Mathematical Physics, (2017), 59(2), 022109.

Faria Da Veiga P.A., Ioriatti L., O’Carroll M. Energy-momentum spectrum of some two-particle lattice Schro¨dinger Hamiltonians. Physical Review E, (2002), 66 , 016130.

Kholmatov Sh.Yu., Lakaev S. N., Almuratov F. Bound states of discrete Schro¨dinger operators on one and two dimensional lattices. J. Math. Anal. Appl.,(2021), 503(1), 125280.

Lakaev S.N., Abdukhakimov S.Kh. Threshold effects in a two-fermion system on an optical lattice. Theoret. and Math. Phys., (2020), 203(2), P. 251–268

Lakaev S.N., Kholmatov Sh.Yu., Khamidov Sh.I. Bose-Hubbard model with onsite and nearest-neighbor interactions; exactly solvable case. J. Phys. A: Math. Theor.,(2021), 54, 245201.

Lakaev S. N., Dell’Antonio G., Khalkhuzhaev A. Existence of an isolated band in a system of three particles in an optical lattice. J. Phys. A: Math. Theor., (2016), 49(52) P. 15.

Lakaev S.N., Lakaev Sh.S. The existence of bound states in a system of three particles in an optical lattice. J.Phys. A: Math. Theor., (2017), 50, 335202.

Lakaev S.N., Motovilov A.K., Abdukhakimov S.Kh. Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions. J. Phys. A: Math., (2023), 56, 315202.

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Published

2026-04-22

Conference Proceedings Volume

Vol. 1 No. 1 (2026): RAQAMLI JAMIYATDA FAN VA TA’LIMNING INTEGRATSIYALALASHUVI: MATEMATIKA, IQTISOD, PEDAGOGIKA VA PSIXOLOGIYANING YANGI YONDASHUVLARI mavzusidagi Respublika ilmiy-texnik anjuman (2026-yil 20-21-aprel)

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How to Cite

Abdukhakimov, S., & Khasanov, A. (2026). DISCRETE SPECTRUM OF TWO-FERMION SCHRÖDINGER OPERATORS ON 3D LATTICES. Scientific Practical Conference, 1(1), 5-7. http://d-pressa.com/index.php/spc/article/view/747