On eigenvalues and virtual levels of a two-particle Hamiltonian on a d-dimensional lattice
https://doi.org/10.17586/2220-8054-2023-14-3-295-303
Abstract
The two-particle Schrödinger operator h_(k); k 2 Td (where _ > 0, Td is a d-dimensional torus), associated to the Hamiltonian h of the system of two quantum particles moving on a d-dimensional lattice, is considered as a perturbation of free Hamiltonian h0(k) by the certain 3d rank potential operator _v. The existence conditions of eigenvalues and virtual levels of h_(k); are investigated in detail with respect to the particle interaction _ and total quasi-momentum k 2 Td.
Keywords
two-particle Hamiltonian,
invariant subspace,
orthogonal projector,
eigenvalue,
virtual level,
multiplicity of virtual level
Supporting agencies: The authors expresses gratitude to the referee for valuable remarks. The work was supported by the Fundamental Science Foundation of Uzbekistan, Grant No. FZ–20200929224.
About the Authors
M. I. Muminov
Samarkand State University; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences
Uzbekistan
Uzbekistan
Mukhiddin I. Muminov
University blv., 15, Samarkand, 140104
Tashkent, 100174
A. M. Khurramov
Samarkand State University
Uzbekistan
Uzbekistan
Abdimajid M. Khurramov
University blv., 15, Samarkand, 140104
I. N. Bozorov
Samarkand State University; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences
Uzbekistan
Uzbekistan
Islom N. Bozorov
University blv., 15, Samarkand, 140104
Tashkent, 100174
References
1. Bloch I., Dalibard J., and Nascimbene S. Quantum simulations with ultracold quantum gases, Nature Physics, 2012, 8, P. 267–276.
2. Jaksch D., Zoller P. The cold atom Hubbard toolbox, Ann. Phys., 2005, 315, P. 52–79.
3. Lewenstein M., Sanpera A., Ahufinger V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems, Oxford University Press, Oxford, 2012.
4. Gullans M., Tiecke T.G., Chang D.E., Feist J., Thompson J.D., Cirac J.I., Zoller P., Lukin M.D. Nanoplasmonic Lattices for Ultracold Atoms. Phys. Rev. Lett., 2012, 109, P. 235309.
5. Hecht E. Optics. Addison-Wesley, Reading, MA. 1998.
6. Murphy B., Hau L.V. Electro-optical nanotraps for neutral atoms. Phys. Rev. Lett., 2009, 102, P. 033003.
7. N.P. de Leon, Lukin M.D., and Park H. Quantum plasmonic circuits. IEEE J. Sel. Top. Quantum Electron., 2012, 18, P. 1781–1791.
8. Bagmutov A.S., Popov I.Y. Window-coupled nanolayers: window shape influence on one-particle and two-particle eigenstates. Nanosystems: Physics, Chemistry, Mathematics, 2020, 11(6), P. 636–641.
9. Hiroshima F., Sasaki I., Shirai T., Suzuki A. Note on the spectrum of discrete Schr¨odinger operators. Journal of Math-for-Industry, 2012, 4, P. 105–108.
10. Higuchi Y., Matsumot T., Ogurisu O. On the spectrum of a discrete Laplacian on Z with finitely supported potential. Linear and Multilinear Algebra, 2011, 8, P. 917–927.
11. Albeverio S., Lakaev S.N., Makarov K.A., Muminov Z.I. The Threshold effects for the two-particle Hamiltonians. Communications in Mathematical Physics, 2006, 262, P. 91–115.
12. Muminov M.I. Positivity of the two-particle Hamiltonian on a lattice. Theoret. and Math. Phys., 2007, 153(3), P. 1671–1676.
13. Muminov M.E., Khurramov A.M. Spectral properties of a two-particle Hamiltonian on a lattice. Theor. Math. Phys., 2013, 177(3), P. 482–496.
14. 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., 2014, 180(3), P. 329–341.
15. Bozorov I.N., Khurramov A.M. On the number of eigenvalues of the lattice model operator in one-dimensional case. Lobachevskii Journal of Mathematics, 2022, 43(2), P. 353–365.
16. Muminov M.E., Khurramov A.M. Spectral properties of two particle Hamiltonian on one-dimensional lattice. Ufa Math. Jour., 2014, 6(4), P. 99–107.
17. Muminov M.I., Khurramov A.M. Spectral properties of a two-particle hamiltonian on a d-dimensional lattice. Nanosystems: Physics, Chemistry, Mathematics, 2016, 7(5), P. 880–887.
18. Lakaev S.N., Bozorov I.N. The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice. Theor. Math. Phys., 2009, 158(3), P. 360–376.
19. Bozorov I.N., Khamidov Sh.I., Lakaev S.N. The number and location of eigenvalues of the two particle discrete Schr¨odinger operators. Lobachevskii Journal of Mathematics, 2022, 43(11), P. 3079–3090.
20. Muminov M.I., Khurramov A.M. On compact distribution of two-particle Schr¨odinger operator on a lattice. Russian Math. (Iz. VUZ), 2015, 59(6), P. 18–22.
21. Muminov M.I., Khurramov A.M., Bozorov I.N. Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice. Nanosystems: Phys. Chem. Math., 2022, 13(3), P. 237–244.
22. Reed M., Simon B. Methods of modern Mathematical Physics. Vol.4. Analysis of Operators, Academic Press, London. 1980. 404 p.
Review
For citations:
Muminov M.I., Khurramov A.M., Bozorov I.N. On eigenvalues and virtual levels of a two-particle Hamiltonian on a d-dimensional lattice. Nanosystems: Physics, Chemistry, Mathematics. 2023;14(3):295-303. https://doi.org/10.17586/2220-8054-2023-14-3-295-303
Views:
4
Email this article 