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Diagonalizing Bose Gases in the Gross-Pitaevskii Regime and Beyond

Morris Brooks

TL;DR

This work develops an algebraic, Bogoliubov-type diagonalization framework for dilute Bose gases in the GP regime and beyond. By combining a Feshbach–Schur renormalization of the two-body interaction with a many-body transformation, it obtains a block-diagonal Hamiltonian that yields a rigorous lower bound matching the GP energy plus the Lee-Huang-Yang (LHY) correction, and it characterizes the low-lying excitation spectrum. The analysis extends to scaling exponents $0\le \kappa<\tfrac{1}{8}$, establishing a sharp link between the many-body spectrum and the Bogoliubov predictions up to $N^{-\tau}$ corrections, and provides an improved upper bound on the ground-state energy that includes a logarithmic $\log N / N$ term. Central to the approach are careful error controls of Bogoliubov-type terms, a priori condensation estimates, and the construction of trial states that annihilate the relevant quasiparticle operators, enabling precise energy comparisons.

Abstract

We present a novel approach to the Bogoliubov theory of dilute Bose gases, allowing for an elementary derivation of the celebrated Lee-Huang-Yang formula in the Gross-Pitaevskii regime. Furthermore, we identify the low lying excitation spectrum beyond the Gross-Pitaevskii scaling, extending a recent result [3] to significantly more singular scaling regimes. Finally, we provide an upper bound on the ground state energy in the Gross-Pitaevskii regime that captures the correct expected order of magnitude beyond the Lee-Huang-Yang formula.

Diagonalizing Bose Gases in the Gross-Pitaevskii Regime and Beyond

TL;DR

This work develops an algebraic, Bogoliubov-type diagonalization framework for dilute Bose gases in the GP regime and beyond. By combining a Feshbach–Schur renormalization of the two-body interaction with a many-body transformation, it obtains a block-diagonal Hamiltonian that yields a rigorous lower bound matching the GP energy plus the Lee-Huang-Yang (LHY) correction, and it characterizes the low-lying excitation spectrum. The analysis extends to scaling exponents , establishing a sharp link between the many-body spectrum and the Bogoliubov predictions up to corrections, and provides an improved upper bound on the ground-state energy that includes a logarithmic term. Central to the approach are careful error controls of Bogoliubov-type terms, a priori condensation estimates, and the construction of trial states that annihilate the relevant quasiparticle operators, enabling precise energy comparisons.

Abstract

We present a novel approach to the Bogoliubov theory of dilute Bose gases, allowing for an elementary derivation of the celebrated Lee-Huang-Yang formula in the Gross-Pitaevskii regime. Furthermore, we identify the low lying excitation spectrum beyond the Gross-Pitaevskii scaling, extending a recent result [3] to significantly more singular scaling regimes. Finally, we provide an upper bound on the ground state energy in the Gross-Pitaevskii regime that captures the correct expected order of magnitude beyond the Lee-Huang-Yang formula.
Paper Structure (14 sections, 22 theorems, 188 equations)

This paper contains 14 sections, 22 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. The two-body Problem
  3. The many-body Diagonalization
  4. Proof of Theorem \ref{['Th:GP']}
  5. Beyond the Gross-Pitaevskii Regime
  6. Control of the Error Terms $\mathcal{E}_i$
  7. Proof of Theorem \ref{['Th:Beyond_GP']}
  8. Trial States and their Energy
  9. Annihilation of $\gamma_k d_k+\nu_k d_{-k}^\dagger$
  10. Annihilation of $\psi_{j k}$
  11. Proof of Theorem \ref{['Th:Upper_Bound']}
  12. Coefficients of the renormalized Potential
  13. Additional Error Estimates
  14. A priori Condensation

Key Result

Theorem 1

Let $E_N$ be the ground state energy of the operator $H_{N}:=H_{N,0}$. Then we have where $\mathfrak{a}_N$ is the box scattering length defined in Eq. (Eq:Box_Scattering_Length).

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 31 more