Meromorphic continuation of the -matrix for the operator acting in a cylinder

Author:
Charles I. Goldstein

Journal:
Proc. Amer. Math. Soc. **42** (1974), 555-562

MSC:
Primary 47F05; Secondary 35P25

MathSciNet review:
0355687

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Abstract: Let and *A* denote the selfadjoint operators given by associated with zero boundary conditions in the domains *S* and , respectively, where *S* is a semi-infinite (or infinite) cylinder with arbitrary cross-section in *N*-dimensional Euclidean space and 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, , such that has spectral multiplicity *m* on and there is a unitary -matrix, , of rank *m* corresponding to each , whose elements may be explicitly given. It is now shown that may be meromorphically continued onto the Riemann surface , obtained by making each a branch point of order one, . Furthermore, the poles are shown to correspond to resonant states.

**[1]**Charles Irwin Goldstein,*Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. I*, Trans. Amer. Math. Soc.**135**(1969), 1–31. MR**0234140**, 10.1090/S0002-9947-1969-0234140-4**[2]**Charles Irwin Goldstein,*Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. II. Applications to scattering theory*, Trans. Amer. Math. Soc.**135**(1969), 33–50. MR**0234141**, 10.1090/S0002-9947-1969-0234141-6**[3]**Charles Goldstein,*Analytic perturbations of the operator -Δ*, J. Math. Anal. Appl.**25**(1969), 128–148. MR**0247310****[4]**Peter D. Lax and Ralph S. Phillips,*Scattering theory*, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR**0217440****[5]**Norman Shenk and Dale Thoe,*Eigenfunction expansions and scattering theory for perturbations of -Δ*, Rocky Mountain J. Math.**1**(1971), no. 1, 89–125. MR**0287189****[6]**Stanly Steinberg,*Meromorphic families of compact operators*, Arch. Rational Mech. Anal.**31**(1968/1969), 372–379. MR**0233240****[7]**M. Ribarič and I. Vidav,*Analytic properties of the inverse 𝐴(𝑧)⁻¹ of an analytic linear operator valued function 𝐴(𝑧)*, Arch. Rational Mech. Anal.**32**(1969), 298–310. MR**0236741**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0355687-3

Keywords:
Singularities of the -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