The analytical nature of the Green’s function in the vicinity of a simple pole
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m1Keywords:
Estimate, pole, eigenvalues, integro-differential conditions, unique solution, Laurent series, adjoint operator, eigenfunction, perturbed boundary, value problem, boundary conditions, Green’s function, resolution, Riesz basis, simple zeroAbstract
It is known that the Green function of a boundary value problem is a meromorphic function
of a spectral parameter. When the boundary conditions contain integro-differential terms,
then the meromorphism of the Green’s function of such a problem can also be proved. In
this case, it is possible to write out the structure of the residue at the singular points of the
Green’s function of the boundary value problem with integro-differential perturbations. An
analysis of the structure of the residue allows us to state that the eigenfunction functions of the
original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have
non-smooth eigenfunctions. The degree of non-smoothness of the eigenfunction of the adjoint
operator to an operator with integro-differential boundary conditions is clarified. It is indicated
that even those conjugate to multipoint boundary value problems have non-smooth eigenfunctions.
