Indefinite boundary eigenvalue problems in a Pontrjagin space setting
We study eigenvalue problems \(Fy=\lambda Gy\) consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric \(\lambda\)-linear boundary conditions. The problems we are interested in are non-definite: neither left- nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a Hermitian form associated with the operator \(F\) has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontryagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher-order are given.
uniformly convergent eigenfunction expansions
scientific article; zbMATH DE number 1320609
scientific article