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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Meromorphic continuation of the $\mathcal {L}$-matrix for the operator $-\Delta$ acting in a cylinder
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by Charles I. Goldstein PDF
Proc. Amer. Math. Soc. 42 (1974), 555-562 Request permission

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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Additional Information
  • © Copyright 1974 American Mathematical Society
  • 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