论文标题
在二维中的一类非本地电位的转移矩阵的存在
Existence of the transfer matrix for a class of nonlocal potentials in two dimensions
论文作者
论文摘要
evan灭波是在相互作用空间的区域中呈指数衰减或成倍增长的波。在schrödinger方程定义的潜在散射中，$（ - \ nabla^2+v）ψ= k^2ψ$对于本地电位$ v $，它们以大于一个的维度出现，无论$ v $的详细信息如何。人们忽略evaneScent波对散射过程的贡献的近似值对应于用一定的依赖能量的非本地势$ \ hat {\ mathscr {v}} _ k $替换$ v $。我们在二维中介绍了$ \ hat {\ mathscr {v}} _ k $的固定散射的动态表述，其中散射数据与具有非换档，无界和非机构的Hamiltonian操作员的量子系统的动力学相关。该系统的进化算子确定了一个维度的固定散射矩阵的二维类似物，其中包含有关电势散射特性的信息。在$ V $的相当普遍的条件下，我们建立了演变操作员的Dyson系列扩展的强烈融合，并证明了$ \ hat {\ Mathscr {\ Mathscr {v}} _ k $的转移矩阵的存在，作为$ \ Mathbb {C}^2^2 2^2^2^2 \ otimes l^2（ - k）的密集定义的操作员。
Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrödinger equation, $(-\nabla^2+v)ψ=k^2ψ$ for a local potential $v$, they arise in dimensions greater than one and are present regardless of the details of $v$. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing $v$ with a certain energy-dependent nonlocal potential $\hat{\mathscr{V}}_k$. We present a dynamical formulation of the stationary scattering for $\hat{\mathscr{V}}_k$ in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on $v$, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for $\hat{\mathscr{V}}_k$ as a densely-defined operator acting in $\mathbb{C}^2\otimes L^2(-k,k)$.
