平方反比势下的量子隧穿和焦散线
Quantum Tunneling and Caustics under Inverse Square Potential
摘要：
本文重新研究了在$-\infty<x<\infty$线上具有平方反比势$V(x)=(m{ω^2}/2){x^2}+g/{x^2}$的谐振子的量子化。结果表明，对于$0<g<3{\hbar^2}/(8m)$，系统存在U（2）族不等价量子化，允许量子隧穿通过$x=0$的势垒.在家庭是一个杰出的量化，减少顺利的谐振子$g\to 0$，在对比传统的量化应用于Calogero模型，禁止隧道，并没有这样的限制。隧穿使经典焦散线在量子水平上变得异常，从而导致从$x>0$的一侧复制任意状态到另一侧$x<0$的可能性。
英文摘要：
Quantization of a harmonic oscillator with inverse square potential $V(x)=(m{ω^2}/2){x^2}+g/{x^2}$ on the line $-\infty<x<\infty$ is re-examined. It is shown that, for $0<g<3{\hbar^2}/(8m)$, the system admits a U(2) family of inequivalent quantizations allowing for quantum tunneling through the potenatial barrier at $x=0$. In the family is a distinguished quantization which reduces smoothly to the harmonic oscillator as $g\to 0$, in contrast to the conventional quantization applied to the Calogero model which prohibits the tunneling and has no such limit. The tunneling renders the classical caustics anomalous at the quantum level, leading to the possibility of copying an arbitrary state from one side $x>0$, say, to the other $x<0$.
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