Sign-indefinite second-order differential operators on finite metric graphs (Q2925851)

Operators of the form \(\displaystyle{-\frac{d}{dx}\, \text{sgn} (x) \,\frac{d}{dx}}\) are generalized to finite metric graphs which need not to be compact. All self-adjoint boundary conditions are characterized and self-adjointness in a Krein space is investigated. Self-adjoint extensions are then looked at in terms of indefinite quadratic forms. Using methods from extension theory all self-adjoint realizations are parametrized. Consequently, spectral and scattering properties of the self-adjoint realizations are proved. In particular, explicit formulae for eigenvalues, resonances and resolvents are given and in addition wave operators and the scattering matrix are computed, the latter, in terms of a generalized star product (in certain cases).