Meromorphic continuation of the $\mathcal{L}$-matrix for the operator $-\Delta$ acting in a cylinder
Charles Irwin Goldstein · Proceedings of the American Mathematical Society · 1974
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.