Friedrichs模型共振态的广义本征向量及其Gamov向量
Generalized Eigenvectors for Resonances in the Friedrichs Model and Their Associated Gamov Vectors
摘要：
构造了R上Friedrichs模型的Hamilton H的Gelfand三重态，使得共振态（Livsic矩阵逆的极点）恰好是H的（广义）本征值.相应的本征反线性形式的计算显式。利用波（Moller）算符的波矩阵，计算了未扰动哈密顿量在Schwartz空间S上的本征反线性形式。证明了它们是纯Dirac型的，并且可以用它们对应的Gamov向量来刻画，而Gamov向量是由S与上半平面的Hardy空间的交的约束唯一确定的.同时，这一限制产生了一个截断的广义演化著名的衰减半群的Toeplitz型的正半线上的Hardy空间。即：如果H的广义本征向量是共振的，并且如果重数参数k来自K的由其相应的Dirac型反线性形式唯一确定的子空间，则这些预Gamov向量（衰变半群的本征向量）恰好具有H的广义本征向量的扩张。
英文摘要：
A Gelfand triplet for the Hamiltonian H of the Friedrichs model on R with finite-dimensional multiplicity space K, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen-antilinearforms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen-antilinearforms on the Schwartz space S for the unperturbed Hamiltonian are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector, which is uniquely determined by restriction of S to the intersection of S with the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup of the Toeplitz type for the positive half line on the Hardy space. That is: exactly those pre-Gamov vectors (eigenvectors of the decay semigroup) have an extension to a generalized eigenvector of H if the eigenvalue is a resonance and if the multiplicity parameter k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinearform.
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