光滑散射理论中的多通道方案
A multichannel scheme in smooth scattering theory
摘要：
在假定$A_{j}A_{k}$与$j\neq k$的所有对积满足一定的正则性条件下，发展了自伴算子对$A_{0}=A_{1}\oplus...\oplus A_{N}$和$A=A_{1}+...+A_{N}$的散射理论.粗略地说，这些条件意味着乘积$A_{j}A_{k}$、$j\neq k$可以表示为在算子$A_{0}$的谱表示中具有光滑核的积分算子。证明了算子$A_{0}$和$A$的绝对连续部分是酉等价的.这产生了一个光滑的版本Ismagilov定理已知早期的跟踪类框架。我们还证明了奇异连续谱的运营商$A$是空的，它的特征值可能只积累到“阈值”的运营商$A_{j}$的绝对连续谱。我们的方法依赖于一个系统的预解方程，可以被认为是一个推广的Faddeev方程的三粒子量子系统。
英文摘要：
We develop the scattering theory for a pair of self-adjoint operators $A_{0}=A_{1}\oplus...\oplus A_{N}$ and $A=A_{1}+...+A_{N}$ under the assumption that all pair products $A_{j}A_{k}$ with $j\neq k$ satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products $A_{j}A_{k}$, $j\neq k$, can be represented as integral operators with smooth kernels in the spectral representation of the operator $A_{0}$. We show that the absolutely continuous parts of the operators $A_{0}$ and $A$ are unitarily equivalent. This yields a smooth version of Ismagilov's theorem known earlier in the trace class framework. We also prove that the singular continuous spectrum of the operator $A$ is empty and that its eigenvalues may accumulate only to "thresholds" of the absolutely continuous spectra of the operators $A_{j}$. Our approach relies on a system of resolvent equations which can be considered as a generalization of Faddeev's equations for three particle quantum systems.
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