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Theoretical and Mathematical Physics

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

Mukhiddin Ishkobilovich Muminov

Published Jul 1, 2010• 4 citations
Publisher
DOI
10.1007/s11232-010-0069-4

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The Bose-Hubbard Hamiltonian can be simplified to have only two lattice sites, in which case the system being described is referred to as a dimer. Due to its structure, the hopping term of the dimer Hamiltonian enjoys invariance in a family of subspaces indexed by a whole number kkk, each subspace corresponding to a system of only kkk particles. We have invented an inductive argument using the bosonic canonical commutation relations to find the eigenvalues and eigenvectors of the dimer hopping Hamiltonian in its kkk-particle subspaces. In particular, this Hamiltonian, when restricted to one of the kkk-particle subspaces, is exactly the spin projection operator along the xxx-axis, where the number of particles kkk in the dimer system yields the projection matrix for spin quantum number s=k/2s=k/2s=k/2. Thus, a new method for computing the eigenvalues and eigenvectors of the xxx-axis spin projector has been unearthed. We use the explicit construction to study the dynamics of coherent states induced by the square of the dimer hopping hamiltonian. We find that it generates Schrödinger cat states in the two-site setting.
We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength μ1∈R\mu_1\in\mathbb{R}μ1​∈R , while the interaction between a fermion and one other particle is via an on-site potential with strength μ2∈R{\mu_2\in\mathbb{R}}μ2​∈R . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum K∈T1K\in\mathbb{T}^1K∈T1 . Furthermore, we show that both the bound state fμ1μ2(K; ⋅ , ⋅ )f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)fμ1​μ2​​(K;⋅,⋅) and its corresponding eigenvalue Eμ1μ2(K)E_{\mu_1\mu_2}(K)Eμ1​μ2​​(K) depend holomorphically on the quasimomentum.