An eigenfunction expansion for a nonselfadjoint, interior point boundary value problem
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 by Allan M. Krall PDF
 Trans. Amer. Math. Soc. 170 (1972), 137147 Request permission
Abstract:
Under discussion is the vector system $Ly = y’ + Py,\sum \nolimits _{j = 0}^\infty {{A_j}y({t_j}) = 0}$, where $\sum \nolimits _{j = 0}^\infty {A < \infty }$. The eigenvalues for the system are known to be countable and approach $\infty$ in the complex plane in a series of welldefined vertical steps. For each eigenvalue there exists an eigenmanifold, generated by the residue of the Green’s function. Further, since the Green’s function vanishes near $\infty$ in the complex plane when the path toward $\infty$ is horizontal, the Green’s function can be expressed as a series of its residues. This in turn leads to two eigenfunction expansions, one for elements in the domain of the original system, another for elements in the domain of the adjoint system.References

G. D. Birkhoff and R. E. Langer, The boundary problems and developments associated with a system of ordinary differential equations of the first order, Proc. Amer. Acad. Arts Sci. 58 (1923), 51128.
 Robert Neff Bryan, A nonhomogeneous linear differential system with interface conditions, Proc. Amer. Math. Soc. 22 (1969), 270–276. MR 241739, DOI 10.1090/S00029939196902417393
 Randal H. Cole, The expansion problem with boundary conditions at a finite set of points, Canadian J. Math. 13 (1961), 462–479. MR 123768, DOI 10.4153/CJM19610395
 Gary B. Green and A. M. Krall, Linear differential systems with infinitely many boundary points, Ann. Mat. Pura Appl. (4) 91 (1972), 53–67. MR 316805, DOI 10.1007/BF02428813
 A. Halanay and A. Moro, A boundary value problem and its adjoint, Ann. Mat. Pura Appl. (4) 79 (1968), 399–411. MR 234052, DOI 10.1007/BF02415186
 Allan M. Krall, Differentialboundary operators, Trans. Amer. Math. Soc. 154 (1971), 429–458. MR 271445, DOI 10.1090/S00029947197102714454 —, Selfadjoint boundary value problems with infinitely many boundary points, Ann. Mat. Pura Appl. (to appear).
 D. H. Tucker, Boundary value problems for linear differential systems, SIAM J. Appl. Math. 17 (1969), 769–783. MR 252741, DOI 10.1137/0117069
Additional Information
 © Copyright 1972 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 170 (1972), 137147
 MSC: Primary 34B25; Secondary 47E05
 DOI: https://doi.org/10.1090/S00029947197203119853
 MathSciNet review: 0311985