离散亚稳态扩散的谱分析
Spectral Analysis of Discrete Metastable Diffusions
AI总结：
该研究分析离散薛定谔算符（离散Witten拉普拉斯算子）在半经典区域的谱底，证明指数小特征值与多阱能量景观的局部极小值一一对应，还计算了双稳态下特征值的渐近分裂，自洽推导出亚稳态隧穿时间的Eyring-Kramers公式。
摘要：
我们研究定义在ℓ²(εℤᵈ)上的离散薛定谔算符$H_\varepsilon= -\varepsilon^2Δ_\varepsilon + V_\varepsilon$，其中ε>0是小参数，势函数$V_\varepsilon$由ℝᵈ上的多阱能量景观f构造。该算符可视为ℝᵈ上半经典Witten拉普拉斯算子的离散对应物，它与εℤᵈ上一个扩散过程的生成元酉等价，且该扩散过程关于玻尔兹曼权重exp(-f/ε)满足细致平衡条件。这类扩散表现出亚稳态行为，常见于统计力学的无序平均场模型中。我们分析了半经典区域ε≪1下$H_\varepsilon$的谱底，证明指数小特征值与f的局部极小值一一对应。随后我们更详细地分析了双稳态情形，计算了两个指数小特征值之间的精确渐近分裂。通过对离散Witten拉普拉斯算子的纯谱理论分析，我们以自洽的方式重新推导出了对应随机过程亚稳态隧穿时间的Eyring-Kramers公式。
英文摘要：
We consider a discrete Schrödinger operator $ H_\varepsilon= -\varepsilon^2Δ_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $\varepsilon>0$ is a small parameter and the potential $V_\varepsilon$ is defined in terms of a multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. It is unitarily equivalent to the generator of a diffusion on $\varepsilon \mathbb Z^d$, satisfying the detailed balance condition with respect to the Boltzmann weight $\exp{(-f/\varepsilon)}$. These type of diffusions exhibit metastable behaviour and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$ and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of $f$. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring-Kramers formula for the metastable tunneling time of the underlying stochastic process.
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