Analyticity of resolvents of elliptic operators on quantum graphs with small edges

TL;DR

This paper studies how the resolvent of elliptic operators on quantum graphs with small edges depends analytically on the small parameter, enabling precise approximations and revealing effects like Robin boundary conditions.

Contribution

It introduces a method to expand the resolvent analytically in the small parameter and shows how higher-order terms influence the limiting operator, extending previous convergence results.

Findings

Resolvent admits a Taylor-like series in the small parameter.

Higher-order terms can induce Robin boundary conditions at vertices.

The approach generalizes to non-analytic and more complex geometries.

Abstract

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator $\mathcal{H}_\varepsilon$ with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in $\varepsilon$. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of $\mathcal{H}_\varepsilon$ are analytic in $\varepsilon$. This allows us to represent the resolvent of $\mathcal{H}_\varepsilon$ by an uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of $\varepsilon$. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in $\varepsilon$-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic $\varepsilon$-dependence of the coefficients in the differential expression and vertex conditions.