论文标题
准确可解决的分段分析双重井潜在$ v_ {d}（x）= min [（x+d）^2，（x-d）^2] $及其双单井潜在$ v_ {s}（x）= max [（x+d）^2，（x-d）^2，（x-d）^2]
Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$
论文作者
论文摘要
通过将两个谐波振荡器电位$ x^2 $与分离$ 2D $并排放置，可以获得两个确切的可解决的分段分析量子系统，并获得了免费参数$ d> 0 $。由于镜子对称性，他们的均值和奇特奇偶校区的特征值$ e $被确定为汇合的超几何功能的某些组合的零，$ {} _ 1f_1 _1f_1 $ d $ $ d $ and $ e $，它们是$ v_ {d} $ v_ {d} $ and $ v_和$ v_ {s} $ concoment。特征函数是$ {} _ 1f_1 $的分段平方集成组合，所谓的$ u $ functions。通过比较分离$ d $的各种值的特征值和特征函数，生动的图片展开，显示了两个井之间的隧道效应。
By putting two harmonic oscillator potential $x^2$ side by side with a separation $2d$, two exactly solvable piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Due to the mirror symmetry, their eigenvalues $E$ for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function ${}_1F_1$ of $d$ and $E$, which are common to $V_{D}$ and $V_{S}$ but in two different branches. The eigenfunctions are the piecewise square integrable combinations of ${}_1F_1$, the so called $U$ functions. By comparing the eigenvalues and eigenfunctions for various values of the separation $d$, vivid pictures unfold showing the tunneling effects between the two wells.
