We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials U(x) ∼ x ^m , m = 2n ⩾ 2. This is paralleled by a transformation of each mth diffusion generator L = DΔ + b(x)∇, and likewise the related Fokker–Planck operator L* = DΔ − ∇[b(x) ⋅], into the affiliated Schrdinger one . Upon a proper adjustment of operator domains, the dynamics is set by semigroups exp(tL), exp(tL*) and exp(−tĤ), with t ⩾ 0. The Feynman–Kac integral kernel of exp(−tĤ) is the major building block of the relaxation process transition probability density, from which L and L* actually follow. The spectral ‘closeness’ of the pertinent Ĥ and the Neumann Laplacian in the interval is analyzed for m even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the m-family of operators Ĥ with a priori chosen , when Ĥ becomes spectrally ‘close’ to the Dirichlet Laplacian for large m. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for Ĥ with a bistable-looking potential has been addressed.

我们调查收敛签名扩散过程的线路上的序列，在保守的力场从超谐波电位词干Ù（X）〜X ^米，米= 2 Ñ ⩾2.这是由每个变换并联米个扩散发生器大号= d Δ+ b（X）∇，同样，相关的福克-普朗克操作者大号* = d Δ - ∇[ b（X）⋅]，进附属薛定谔一个 。当运营商域的适当调整，动力学是由半群EXP（设置TL），实验值（TL *）和EXP（ - TH），以吨⩾0。费曼-KAC的EXP积分核（ - TH）是L和L *实际上是从中得出的弛豫过程过渡概率密度的主要组成部分。分析了相关Ĥ和Neumann Laplacian在区间中的光谱“接近度”，其中m均匀且较大。随着讨论的副产物，我们给出一个类似的亲和力的详细描述，在条款米-家庭运营商的Ĥ在选择先验的情况下，当m在光谱上变得“接近”大m的Dirichlet Laplacian时。为了完善，已经解决了一个问题，即对于看起来具有双稳态潜力的the，不存在负特征值。