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A two-boson lattice Hamiltonian with interactions up to next-neighboring sites

S. N. Lakaev, A. K. Motovilov, M. O. Akhmadova

Abstract

A system of two identical spinless bosons on the two-dimensional lattice is considered under the assumption that on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes $γ$, $λ$, and $μ$, respectively. A partition of the $(γ,λ,μ)$-space into connected components is established such that, in each connected component, the two-boson Schroedinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover, for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schrödinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.

A two-boson lattice Hamiltonian with interactions up to next-neighboring sites

Abstract

A system of two identical spinless bosons on the two-dimensional lattice is considered under the assumption that on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes , , and , respectively. A partition of the -space into connected components is established such that, in each connected component, the two-boson Schroedinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover, for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schrödinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.

Paper Structure

This paper contains 16 sections, 20 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian of a two-boson system on the lattice $\mathbb{Z}^2$ and its basic properties
  3. The position-space representation
  4. The quasimomentum-space representation.
  5. The Floquet-Bloch decomposition of $\mathbb{H}_{\gamma\lambda\mu}$ and fiber Schrödinger operators
  6. Essential spectrum of the fiber Schrödinger operators
  7. Relationship between the discrete spectra of $H_{\gamma\lambda\mu}(K)$ for $K\neq 0$ and $H_{\gamma\lambda\mu}(0)$
  8. Reducing subspaces of the fiber Schrödinger operators $H_{\gamma\lambda\mu}(0)$
  9. Main results: The discrete spectrum of the two-boson fiber Hamiltonians
  10. The discrete spectrum of ${H}_{\mu}^\mathrm{oos}(0)$.
  11. The discrete spectrum of ${H}_{\lambda\mu}^\mathrm{ea}(0)$.
  12. The discrete spectrum of ${H}_{\gamma\lambda\mu}^\mathrm{ees}(0)$.
  13. The total discrete spectrum of $H_{\gamma\lambda\mu}(K)$
  14. Auxiliary statements
  15. The Lippmann--Schwinger operator
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $n\ge1$ and $i\in\{1,2\}.$ For each fixed $K_j\in\mathbb{T},$$j\in\{1,2\}\setminus\{i\},$ the map is non-increasing in $(-\pi,0]$ and non-decreasing in $[0,\pi]$. Similarly, for every fixed $K_j\in\mathbb{T},$$j\in\{1,2\}\setminus\{i\},$ the map is non-increasing in $(-\pi,0]$ and non-decreasing in $[0,\pi]$.

Figures (2)

  • Figure 1: A schematic location of the sets ${\mathcal{A}}_{j}^\pm$, $j=0,1,2,$ defined in \ref{['six_sets_p']}.
  • Figure 2: (A) Plots of the curves $\tau^-_j$, $j=1,2,3$, partitioning the domain of the function $\gamma^{-}$ into the parts $D^-_\alpha$, $\alpha=1,2,3,4$. (B) Plots of the curves $\tau^+_j$, $j=1,2,3$, partitioning the domain of the function $\gamma^{+}$ into the parts $D^+_\alpha$, $\alpha=1,2,3,4$.

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2: LHA:2024, Theorem 3.3
  • Lemma 3.3
  • ...and 23 more