Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analyticity of resolvents of elliptic operators on quantum graphs with
  small edges   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.

3
The categories of this paper are: physics
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas