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Bound states of 2+1 fermionic trimers on lattice at strong couplings

Janikul Abdullaev, Ahmad Khalkhuzhaev, Shokhrukh Yu. Kholmatov

TL;DR

The paper analyzes bound states of a 2+1 fermionic trimer on a 3D lattice with zero-range interactions by reducing the three-body spectral problem to a Birman-Schwinger-type operator. A fiber decomposition via the Brillouin zone yields precise descriptions of the essential spectrum, consisting of two branches from the two-particle and three-particle channels, and reveals parameter-dependent thresholds in the mass ratio $\gamma$ that govern the presence or absence of discrete eigenvalues below the essential spectrum or within the spectral gap. The main tool is a careful large-$\lambda$ analysis, decomposing the Birman-Schwinger operator into a finite-rank principal part and a small-norm residual, enabling rigorous counting and parity analysis of bound states, particularly at the symmetric point $K=0$. The results demonstrate that bound states on the lattice emerge only when the fermion mass is sufficiently large relative to the third particle, with explicit thresholds and multiplicities; in the gap, bound states can exist without a ground state, and near the two-particle edge the spectrum can contain several eigenvalues depending on $\gamma$.

Abstract

In this paper, we investigate the bound states of $2+1$ fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator $H_{γ,λ}(K),$ focusing on energies below the continuum and within its gap. Depending on the quasi-momentum $K,$ we show that if the mass ratio $γ>0$ between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling $λ>0.$ Conversely, if $γ$ exceeds this threshold, $H_{γ,λ}(K)$ admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as $λ\to+\infty.$ For $K=0,$ the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.

Bound states of 2+1 fermionic trimers on lattice at strong couplings

TL;DR

The paper analyzes bound states of a 2+1 fermionic trimer on a 3D lattice with zero-range interactions by reducing the three-body spectral problem to a Birman-Schwinger-type operator. A fiber decomposition via the Brillouin zone yields precise descriptions of the essential spectrum, consisting of two branches from the two-particle and three-particle channels, and reveals parameter-dependent thresholds in the mass ratio that govern the presence or absence of discrete eigenvalues below the essential spectrum or within the spectral gap. The main tool is a careful large- analysis, decomposing the Birman-Schwinger operator into a finite-rank principal part and a small-norm residual, enabling rigorous counting and parity analysis of bound states, particularly at the symmetric point . The results demonstrate that bound states on the lattice emerge only when the fermion mass is sufficiently large relative to the third particle, with explicit thresholds and multiplicities; in the gap, bound states can exist without a ground state, and near the two-particle edge the spectrum can contain several eigenvalues depending on .

Abstract

In this paper, we investigate the bound states of fermionic trimers on a three-dimensional lattice at strong coupling. Specifically, we analyze the discrete spectrum of the associated three-body discrete Schrödinger operator focusing on energies below the continuum and within its gap. Depending on the quasi-momentum we show that if the mass ratio between the identical fermions and the third particle is below a certain threshold, the operator lacks a discrete spectrum below the essential spectrum for sufficiently large coupling Conversely, if exceeds this threshold, admits at least one eigenvalue below the essential spectrum. Similar phenomena are observed in the neighborhood of the two-particle branch of the essential spectrum, which resides within the gap and grows sublinearly as For the mass ratio thresholds are explicitly calculated and it turns out that, for certain intermediate mass ratios and large couplings, bound states emerge within the gap, although ground states are absent.

Paper Structure

This paper contains 19 sections, 18 theorems, 255 equations.

Table of Contents

  1. Introduction and main results
  2. Background
  3. The model
  4. Main results
  5. Some comments on the main results
  6. A main strategy of proofs
  7. The structure of the paper
  8. Essential spectrum. Spectral properties of two-particle operators
  9. Channel operators
  10. Some comments on the spectrum of $h_{\gamma,\lambda}(K,q)$
  11. Essential spectrum of $H_{\gamma,\lambda}(K)$
  12. Birman-Schwinger-type principle
  13. Behaviour of $\mathcal{B}_{\gamma,K}(z,\lambda)$ for large $\lambda$ and $|z|$
  14. Discrete spectrum of $H_{\gamma,\lambda}(K)$
  15. On the discrete spectrum of $H_{\gamma,\lambda}(0)$
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $K\in\mathbb{T}^3.$ There exist $\gamma_1:=\gamma_1(K)>0$ and $\widetilde{\gamma}_1:=\widetilde{\gamma}_1(K)>0$ with the following property.

Theorems & Definitions (37)

  • Remark 1.1
  • Theorem 1.2: Eigenvalues of $H_{\gamma,\lambda}(K)$ below the essential spectrum
  • Theorem 1.3: Eigenvalues of $H_{\gamma,\lambda}(K)$ in the gap
  • Corollary 1.4: Satellite eigenvalues of two particle branch of essential spectrum
  • Theorem 1.5
  • Theorem 1.6: Eigenvalues in the gap
  • Remark 2.1
  • Lemma 2.2: Properties of $\widehat{z}_{\gamma,\lambda}$
  • proof
  • Remark 2.3
  • ...and 27 more