Scattering zippers and their spectral theory (Q1943724)

The authors consider a scattering zipper which represents a system obtained by concatenation of scattering events with equal number of incoming and out-going channels. The associated operator \(\mathbb{U}_N\) acting on \(\ell^2(\{1,\dots,N\},\mathbb{C}^L)\) is defined by \[ \mathbb{U}_N=\mathbb{V}_N\mathbb{W}_N, \] where \(N\) is either even or infinite and \[ \mathbb{V}_N=\mathrm{diag}\{S_2,S_4,\dots,S_N\}, \quad \mathbb{W}_N=\mathrm{diag}\{U,S_3,S_5,\dots,S_{N-1},V\} \] with \(L\times L\) unitary matrices \(U,V\) and a sequence \((S_n)_{n=2,3,\dots}\) of \(2L\times 2L\) unitary matrices such that the \(L\times L\) right upper block of each \(S_n\) is invertible. Several properties of this operator are established throughout the paper which consists of nine sections and an appendix. In particular, the resolvents are studied in Section 4 (see Theorem 1), Weyl theory is considered in Section 5 (see Theorems 2 and 3), semi-infinite operators \(\mathbb{U}\) are treated in Section 6 (see Theorem 4), eigenvalues are investigated in Sections 7 and 8 (see Theorems 5 and 6), and the spectrum of infinite periodic scattering zippers is characterized in Section 9 (see Theorem 7). Moreover, from these results one can conclude that the scattering zippers are the unitary analogue of Jacobi matrices with matrix entries.