AI 中文总结
该研究针对ℝⁿ上对称双势阱的半经典薛定谔算子，严格实现稀薄瞬子图像，通过热核迹分解证明双势阱瞬子服从泊松分布，推导得到本征值分裂的瞬子展开式。
AI 中文摘要
我们针对ℝⁿ上具有对称双势阱的半经典薛定谔算子，给出了稀薄瞬子图像的严格实现。利用局域化的Feynman-Kac表示，我们根据布朗桥在两个势阱收缩邻域之间的穿越次数，对热核迹进行分解。我们将一次穿越的权重与跃迁系数ρ_λ对应，ρ_λ=∫_{∂Ω}(∇\bar{φ_{λ,0}^Ω}φ_{λ,0}^{-Ω}-\bar{φ_{λ,0}^Ω}∇φ_{λ,0}^{-Ω})·ν。在指数长的时间尺度β=N/|ρ_λ|上，对每个固定的N>0，穿越次数收敛于均值为N的泊松随机变量。我们确定-1/λ log|ρ_λ|→S(d,-d)，并得到E₁(λ)-E₀(λ)=2|ρ_λ|(1+o(1))。因此，双势阱本征值分裂的熟知瞬子展开式直接由热核迹的因式分解得到。
英文摘要
We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman--Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient $ρ_λ$, $\displaystyle ρ_λ= \int_{\partialΩ} \left( \nabla\overline{φ_{λ,0}^Ω}\, φ_{λ,0}^{-Ω} - \overline{φ_{λ,0}^Ω}\, \nablaφ_{λ,0}^{-Ω} \right)\cdotν. $ On the exponentially long time scale $β=N/\lvertρ_λ\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}λ\log\lvertρ_λ\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(λ)-E_0(λ) = 2\lvertρ_λ\rvert\left(1+o(1)\right). $ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.
Comments34 pages, no figures
