On the discrete spectrum of the Schr\"odinger operator using the 2+1 fermionic trimer on the lattice

Ahmad M. Khalkhuzhaev ; Islom. A. Khujamiyorov

Nanosistemy: fizika, himiâ, matematika, Tome 14 (2023) no. 5, pp. 518-529

Résumé

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 $m=1/\gamma1$), 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 $\mu$.

Détail

Keywords: Schrödinger operator, Hamiltonian, contact potential, eigenvalue, quasi-momentum, invariant subspace, Faddeev operator.
Mots-clés : fermion