A Conic Manifold Perspective of Elliptic Operators on Graphs
نویسندگان
چکیده
The analysis of differential operators on graphs is an area of current interest with a long tradition (cf. [1, 7, 8, 11], in particular the survey article [9]). In this paper we adopt the point of view that a graph is a one-dimensional manifold with conical singularities, and that a differential operator on it with smooth coefficients and regular singular points at most at the endpoints of the edges is a cone differential operator. Among these, the simplest ones are second order operators whose coefficients are smooth up to the boundary, except for the potential term, which is permitted to have Coulomb type singularities. This class of operators is interesting not just intrinsically but also because it is already one for which the notion of boundary values is not trivial. The aim of the present paper is to analyze in detail the existence of sectors of minimal growth for the latter class of operators, the main point being that this can be done explicitly and in simple terms. We follow the approach developed in [4, 5, 6] on general elliptic cone operators. At the end of the paper we shall indicate how the general situation on a graph can be analyzed with the same tools. Recall that a closed sector
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On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators
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