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MATHEMATICS
On the discrete spectrum of the Schrödinger operator using the 2+1 fermionic trimer on the lattice
Ahmad M. Khalkhuzhaeva, Islom. A. Khujamiyorovb a Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan, Uzbekistan
b Samarkand State University, Samarkand, Uzbekistan
Abstract:
We consider the three-particle discrete Schrödinger operator $H_{\mu,\gamma}(\mathbf{K})$, $\mathbf{K}\in\mathbb{T}^3$, associated with the three-particle Hamiltonian (two of them are fermions with mass 1 and one of them is arbitrary with mass $m=1/\gamma<1$), interacting via pair of repulsive contact potentials $\mu>0$ on a three-dimensional lattice $\mathbb{Z}^3$. It is proved that there are critical values of mass ratios $\gamma=\gamma_1$ and $\gamma=\gamma_2$ such that if $\gamma\in(0,\gamma_1)$, then the operator $H_{\mu,\gamma}(0)$ has no eigenvalues. If $\gamma\in(\gamma_1,\gamma_2)$, then the operator $H_{\mu,\gamma}(0)$ has a unique eigenvalue; if $\gamma>\gamma_2$, then the operator $H_{\mu,\gamma}(0)$ has three eigenvalues lying to the right of the essential spectrum for all sufficiently large values of the interaction energy $\mu$.
Keywords:
Schrödinger operator, Hamiltonian, contact potential, fermion, eigenvalue, quasi-momentum, invariant subspace, Faddeev operator.
Received: 03.07.2023 Revised: 13.09.2023 Accepted: 14.09.2023
Citation:
Ahmad M. Khalkhuzhaev, Islom. A. Khujamiyorov, “On the discrete spectrum of the Schrödinger operator using the 2+1 fermionic trimer on the lattice”, Nanosystems: Physics, Chemistry, Mathematics, 14:5 (2023), 518–529
Linking options:
https://www.mathnet.ru/eng/nano1217 https://www.mathnet.ru/eng/nano/v14/i5/p518
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