Abstract
In this article, we present a three-particle lattice model Hamiltonian \({{H}_{{\mu ,\lambda }}}\), \(\mu ,\lambda > 0\) by making use nonlocal potential. The Hamiltonian under consideration acts as a tensor sum of two Friedrichs models \({{h}_{{\mu ,\lambda }}}\) which comprises a rank 2 perturbation associated with a system of three quantum particles on a d-dimensional lattice. The current study investigates the number of eigenvalues associated with the Hamiltonian. Furthermore, we provide the suitable conditions on the existence of eigenvalues localized inside, in the gap and below the bottom of the essential spectrum of \({{H}_{{\mu ,\lambda }}}\).
REFERENCES
G. M. Graf and D. Schenker, “2-Magnon scattering in the Heisenberg model,” Ann. Inst. Henri Poincaré Phys. Théor. 67, 91–107 (1997).
P. A. Faria da Veiga, L. Ioriatti, and M. O’Carroll, “Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians,” Phys. Rev. E 66, 16130 (2002). https://doi.org/10.1103/PhysRevE.66.016130
D. Mattis, “The few-body problem on a lattice,” Rev. Mod. Phys. 58, 361–379 (1986). https://doi.org/10.1103/revmodphys.58.361
A. I. Mogilner, “Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: Problems and results,” Adv. Sov. Math. 5, 139–194 (1991). https://doi.org/10.1090/advsov/005/05
V. A. Malyshev and R. A. Minlos, Linear Infinite-Particle Operators, Translations of Mathematical Monographs, Vol. 143 (American Mathematical Society, 1995). https://doi.org/10.1090/mmono/143
S. Albeverio, S. N. Lakaev, and R. Kh. Djumanova, “The essential and discrete spectrum of a model operator associated to a system of three identical quantum particles,” Rep. Math. Phys. 63, 359–380 (2009). https://doi.org/10.1016/s0034-4877(09)00017-2
S. Albeverio, S. N. Lakaev, and Z. I. Muminov, “On the number of eigenvalues of a model operator associated to a system of three-particles on lattices,” Russ. J. Math. Phys. 14, 377–387 (2007). https://doi.org/10.1134/s1061920807040024
T. Kh. Rasulov and R. T. Mukhitdinov, “The finiteness of the discrete spectrum of a model operator associated with a system of three particles on a lattice,” Russ. Math. 58, 52–59 (2014). https://doi.org/10.3103/s1066369x1401006x
V. Heine, “The pseudopotential concept,” in Solid State Physics, Ed. by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1970), Vol. 24, pp. 1–36. https://doi.org/10.1016/S0081-1947(08)60069-7
B. V. Karpenko, V. V. Dyakin, and G. A. Budrina, “Two electrons in Hubbard model,” Phys. Met. Metallogr. 61, 702–706 (1986).
M. É. Muminov, “Expression for the number of eigenvalues of a Friedrichs model,” Math. Notes 82, 67–74 (2007). https://doi.org/10.1134/S0001434607070097
S. Albeverio, S. N. Lakaev, and Z. I. Muminov, “The threshold effects for a family of Friedrichs models under rank one perturbations,” J. Math. Anal. Appl. 330, 1152–1168 (2007). https://doi.org/10.1016/j.jmaa.2006.08.046
S. Albeverio, S. N. Lakaev, and T. H. Rasulov, “On the spectrum of an Hamiltonian in Fock Space. Discrete spectrum asymptotics,” J. Stat. Phys. 127, 191–220 (2007). https://doi.org/10.1007/s10955-006-9240-6
M. I. Muminov, T. H. Rasulov, and N. A. Tosheva, “Analysis of the discrete spectrum of the family of 3 × 3 operator matrices,” Commun. Math. Anal. 23, 17–37 (2020).
T. H. Rasulov and E. B. Dilmurodov, “Infinite number of eigenvalues of 2 × 2 operator matrices: Asymptotic discrete spectrum,” Theor. Math. Phys. 205, 1564–1584 (2019). https://doi.org/10.1134/s0040577920120028
T. H. Rasulov and E. B. Dilmurodov, “Analysis of the spectrum of a 2 × 2 operator matrix. Discrete spectrum asymptotics,” Nanosystems: Phys., Chem., Math. 11, 138–144 (2020). https://doi.org/10.17586/2220-8054-2020-11-2-138-144
T. H. Rasulov and E. B. Dilmurodov, “Infinite number of eigenvalues of 2 × 2 operator matrices: Asymptotic discrete spectrum,” Theor. Math. Phys. 205, 1564–1584 (2020). https://doi.org/10.1134/S0040577920120028
T. H. Rasulov and E. B. Dilmurodov, “Issledovanie chislovoi oblasti znachenii odnoi operatornoi matritsy,” Vestn. Samarsk. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki 2 (35), 50–63 (2014). https://doi.org/10.14498/vsgtu1275
T. H. Rasulov and E. B. Dilmurodov, “Estimates for the bounds of the essential spectrum of a 2 × 2 operator matrix,” Contemp. Math. 1, 170–186 (2020). https://doi.org/10.37256/cm.142020409
M. Reed and B. Simon, Methods of Modern Mathematical Physics: Functional Analysis (Academic, New York, 1979). https://doi.org/10.1016/B978-0-12-585001-8.X5001-6
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Bahronov, B.I., Rasulov, T.H. & Rehman, M. Conditions for the Existence of Eigenvalues of a Three-Particle Lattice Model Hamiltonian. Russ Math. 67, 1–8 (2023). https://doi.org/10.3103/S1066369X23070010
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DOI: https://doi.org/10.3103/S1066369X23070010