论文标题
挤压双层结构的散射数据和绑定状态
Scattering data and bound states of a squeezed double-layer structure
论文作者
论文摘要
作为宽度$ l_1 $和$ l_2 $，研究了一个由两个平行均匀层组成的异质结构，它们之间的距离$ r $同时缩小到零。该问题在一个维度上进行了研究，Schrödinger方程中的挤压潜力由强度$ v_1 $和$ v_2 $给出，具体取决于层厚度。全类函数$ v_1（l_1）$和$ v_2（l_2）$由某些限制特性指定为$ l_1 $，$ l_2 $倾向于零。仅当系统参数上的某些条件$ v_j $，$ l_j $，$ j = 1,2 $和$ r $发生的情况下，散射数据$ a（k）$ a（k）$（k）$（k）$（k）$（k）$的挤压极限才存在。这些条件是由于适当取消差异而出现的。进行了两种取消方法，并得出了系统参数空间中相应的两个共振集。在这些集合之一上，在挤压极限中证明了非平凡的结合状态的存在，包括以Dirac的Delta功能衍生物的形式挤压潜力的特定例子，与对$δ'$ - 类似系统中绑定状态的不存在的不存在的意见相反。详细介绍了从有限系统中有限数量的有限状态在挤压系统中生存的方案。
A heterostructure composed of two parallel homogeneous layers is studied in the limit as their widths $l_1$ and $l_2$, and the distance between them $r$ shrinks to zero simultaneously. The problem is investigated in one dimension and the squeezing potential in the Schrödinger equation is given by the strengths $V_1$ and $V_2$ depending on the layer thickness. A whole class of functions $V_1(l_1)$ and $V_2(l_2)$ is specified by certain limit characteristics as $l_1$ and $l_2$ tend to zero. The squeezing limit of the scattering data $a(k)$ and $b(k)$ derived for the finite system is shown to exist only if some conditions on the system parameters $V_j$, $l_j$, $j=1,2$, and $r$ take place. These conditions appear as a result of an appropriate cancellation of divergences. Two ways of this cancellation are carried out and the corresponding two resonance sets in the system parameter space are derived. On one of these sets, the existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function, contrary to the widespread opinion on the non-existence of bound states in $δ'$-like systems. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.
