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Meromorphic continuation of the $ \mathcal{L}$-matrix for the operator $ -\Delta$ acting in a cylinder


Author: Charles I. Goldstein
Journal: Proc. Amer. Math. Soc. 42 (1974), 555-562
MSC: Primary 47F05; Secondary 35P25
DOI: https://doi.org/10.1090/S0002-9939-1974-0355687-3
MathSciNet review: 0355687
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Abstract: Let $ {A^c}$ and A denote the selfadjoint operators given by $ - \Delta $ associated with zero boundary conditions in the domains S and $ \Omega $, respectively, where S is a semi-infinite (or infinite) cylinder with arbitrary cross-section in N-dimensional Euclidean space $ (N \geqq 2)$ and $ \Omega $ is obtained from S by perturbing a finite portion of the boundary of S. It has been previously shown that there exists a set of intervals, $ {G_m} = [{v_m},{v_{m + 1}}),m = 1,2, \cdots $, such that $ 0 < {v_m} < {v_{m + 1}} < \infty ,{A_0}$ has spectral multiplicity m on $ {G_m}$ and there is a unitary $ \mathcal{S}$-matrix, $ {\mathcal{S}_m}(\lambda )$, of rank m corresponding to each $ {G_m}$, whose elements may be explicitly given. It is now shown that $ {\mathcal{S}_m}(\lambda )$ may be meromorphically continued onto the Riemann surface $ {R_m}$, obtained by making each $ {v_j}$ a branch point of order one, $ j = 1, \cdots $. Furthermore, the poles are shown to correspond to resonant states.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0355687-3
Keywords: Singularities of the $ \mathcal{S}$-matrix, waveguides, limiting absorption principle, radiation condition, infinitely sheeted Riemann surface, meromorphic continuation, Schwarz reflection principle, resonant states
Article copyright: © Copyright 1974 American Mathematical Society