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 $\mu_1\in\mathbb{R}$ , while the interaction between a fermion and one other particle is via an on-site potential with strength ${\mu_2\in\mathbb{R}}$ . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum $K\in\mathbb{T}^1$ . Furthermore, we show that both the bound state $f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)$ and its corresponding eigenvalue $E_{\mu_1\mu_2}(K)$ depend holomorphically on the quasimomentum.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 $$\mu_1\in\mathbb{R}$$ , while the interaction between a fermion and one other particle is via an on-site potential with strength $${\mu_2\in\mathbb{R}}$$ . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum $$K\in\mathbb{T}^1$$ . Furthermore, we show that both the bound state $$f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)$$ and its corresponding eigenvalue $$E_{\mu_1\mu_2}(K)$$ depend holomorphically on the quasimomentum.

Theoretical and Mathematical PhysicsDec 1, 20253 citations

Excitations for lattice ferromagnetic classical spin systems at high temperature: Noneven single-spin distributions

Ricardo S. Schor, Michael O’Carroll

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region $(\ensuremath{\beta}\ensuremath{\ll}1).$ Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter $\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),$ where $\overline{s}=s\ensuremath{-}〈s〉,$ and $〈{s}^{k}〉$ denotes the $k\mathrm{th}$ moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold $2m.$ For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to $2m.$ We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region $(\ensuremath{\beta}\ensuremath{\ll}1).$ Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter $\ensuremath{\alpha}=(〈\overline{s}{}^{4}〉\ensuremath{-}3〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1})/(〈\overline{s}{}^{4}〉\ensuremath{-}〈\overline{s}{}^{2}{〉}^{2}\ensuremath{-}〈\overline{s}{}^{3}{〉}^{2}〈\overline{s}{}^{2}{〉}^{\ensuremath{-}1}),$ where $\overline{s}=s\ensuremath{-}〈s〉,$ and $〈{s}^{k}〉$ denotes the $k\mathrm{th}$ moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold $2m.$ For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to $2m.$

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 20005 citations

Bound states of a system of two fermions on a one-dimensional lattice

J. I. Abdullaev

Theoretical and Mathematical PhysicsApr 1, 200614 citations

Few body bound states in a lattice

Jugal Talukdar, D. Blume

Bulletin of the American Physical SocietyMay 31, 20180 citations

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

\mu_1\in\mathbb{R}

, while the interaction between a fermion and one other particle is via an on-site potential with strength

{\mu_2\in\mathbb{R}}

. We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum

K\in\mathbb{T}^1

. Furthermore, we show that both the bound state

f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)

and its corresponding eigenvalue

E_{\mu_1\mu_2}(K)

depend holomorphically on the quasimomentum.

〈{s}^{k}〉

denotes the

k\mathrm{th}

moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass $m\ensuremath{\sim}\mathrm{ln}\ensuremath{\beta}.$ Assuming $〈{s}^{3}〉\ensuremath{\ne}0$ we show that for $\ensuremath{\alpha}>0,$ \ensuremath{\beta} small, there exists a bound state with mass below the two-particle threshold

2m.

For $\ensuremath{\alpha}<0$ bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to

2m.