Sign-indefinite second order differential operators on finite metric graphs

TL;DR

This paper investigates self-adjoint realizations of sign-indefinite second order differential operators on finite metric graphs, providing a comprehensive parametrization and analyzing their spectral and scattering properties.

Contribution

It extends the theory of such operators to finite metric graphs and offers a detailed spectral and scattering analysis, which was not previously available.

Findings

All self-adjoint realizations are parametrized using extension theory.

Spectral properties of the operators are characterized.

Scattering theory for these operators is developed in detail.

Abstract

The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theory of the self-adjoint realizations are studied in detail.