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本文研究二维晶格上带两两接触相互作用的三玻色子薛定谔算子，得到不同准动量下的束缚态能量、谱间隙，证明基态指数局域化，其结果对少玻色子系统的量子模拟有直接意义。
AI 中文摘要
我们研究二维整数晶格上带有两两接触相互作用的三玻色子薛定谔算子。首先，我们在强耦合极限μ→∞下，得到零总准动量处本征谱下方两个束缚态的精确渐近：基态能量为-3μ + 6 + O(1/μ)，第一激发态能量为-μ + C + O(1/μ)，其中C=4-δ≈3.96458，δ由涉及晶格格林函数的超越方程b₀(δ)=1/(1+δ)确定，对应的谱间隙为2μ + O(1)。其次，利用离散Agmon比较方法和Paley-Wiener定理，我们建立了基态波函数的指数局域化，衰减率α(μ)的对数上界至多为ln(3μ) + O(1/μ)，这反映了晶格色散的有界性；我们还对尖锐渐近率提出了猜想。第三，在准动量π处，宇称对称性得以保留，但二次奇型形式成为正定，其唯一本征值为1/μ量级，且永远不会达到Birman-Schwinger阈值，因此奇子空间仅产生一个虚能级。偶子空间恰好支持一个束缚态，其能量为-2μ + 6 + 8/μ + O(1/μ²)，到本征谱的谱间隙为μ - 2 - 8/μ + O(1/μ²)。从零准动量处的两个束缚态减少到准动量π处的一个束缚态，这一过程保持了总谱流，是晶格特有的现象。我们的方法采用Birman-Schwinger算子的不变子空间分解和Krein-Rutman定理，以保证基态的唯一性和严格正定性。这些结果对光晶格中少玻色子系统的量子模拟具有直接意义。
英文摘要
We study three identical bosons with contact attraction of strength mu on the square lattice at total quasimomentum K = 0 and K = pi. First, at K = 0, for all sufficiently large mu there are exactly two eigenvalues below the essential spectrum. The trimer branch follows from Kato perturbation theory at the simple isolated eigenvalue 3 of the pair-contact operator, uniformly in K; the atom-dimer branch follows from a Feshbach-Schur reduction whose limiting equation coincides with the second Birman-Schwinger (Faddeev) eigenvalue of the finite-rank principal part. The spectral gap between the two branches grows linearly in mu. Second, at K = pi the trimer branch is unchanged up to O(mu^-2), the two-particle threshold equals -mu + 4 exactly, and the gap to it is 2mu - 2 plus lower order corrections. The second bound state disappears because the exchange operator changes sign, S_pi = -S_0: the s-channel eigenvalue that exceeds 1 at K = 0 becomes negative, while the only positive atom-dimer channel at K = pi is the odd p-channel, bounded by 2 - 4/pi < 1. Consequently the operator has exactly one eigenvalue below the threshold minus 36/(mu - 10). Third, the ground state is exponentially localized in the relative coordinates. The decay exponent is positive unconditionally; under a non-cancellation hypothesis it is bounded above by log(3mu) plus lower order terms, while finite-volume computations indicate the slowest tail is the atom-dimer dissociation channel with decay rate one half log(2mu). A dictionary between the principal-part method, Kato theory, and the atom-dimer threshold matrix is given, together with the three-dimensional constants for comparison.
Commentsv3 corrects the Faddeev normalisation in the even sector at K=pi, removes the spurious branch it had produced, refines the atom-dimer characteristic root, sharpens the epistemic status of the decay-rate estimates, and consolidates the abstract and conclusion. 69 pages, 5 figures
