An eigenfunction expansion for a nonselfadjoint, interior point boundary value problem

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 well-defined 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.