Quantum graphs on radially symmetric antitrees

Abstract

We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs admits a lot of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm-Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension.Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, a criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is OE0; 1/.

Self‐adjoint and Markovian extensions of infinite quantum graphs

“…If vol() < ∞, then Markovian extensions of 𝐇 0 form a one-parameter family explicitly given by (6.17). Note that (6.17) looks similar to the description of self-adjoint extensions of the minimal Kirchhoff Laplacian on radially symmetric antitrees obtained recently in [56].…”

“…Example Consider a metric antitree $${\mathcal {G}}= {\mathcal {A}}$$ (see Section 7.1 for definitions) and additionally suppose that $${\mathcal {A}}$$ is radially symmetric , that is, for each $$n\geqslant 0$$, all edges between the combinatorial spheres $$S_n$$ and $$S_{n+1}$$ have the same length. Combining [56, Theorem 4.1] (see also Corollary 7.3) with the fact that antitrees have exactly one graph end, $$\# \mathfrak {C}({\mathcal {A}}) = 1$$, we conclude that $$\begin{equation*} {\rm {n}}_\pm ({\mathbf {H}}_0) = \# \mathfrak {C}_0({\mathcal {G}}) = {\begin{cases} 1, &\text{if } {\rm {vol}}({\mathcal {A}}) &lt; \infty , \\ 0, & \text{if } {\rm {vol}}({\mathcal {A}}) = \infty . \end{cases}} \end{equation*}$$In particular, $${\mathbf {H}}$$ is self‐adjoint if and only if $${\rm {vol}}({\mathcal {A}}) = \infty$$.…”

“…The latter in particular implies the following statement (cf. [56, Theorem 4.1]). Corollary If $${\mathcal {A}}$$ is a radial antitree with finite total volume, then $${\rm {n}}_\pm ({\mathbf {H}}_0) = 1$$.…”

Szego condition, scattering, and vibration of Krein strings

“…However, we expect that our methods are fairly universal and can be used to consider other operators. Also, we believe that some of our results can find applications in the theory of Gaussian stochastic processes (see, e.g., [25] for background), wave equations with fractal Laplacians [13], and scattering theory on quantum graphs [44].…”

“…This characterization was obtained in [5]. Below, we give somewhat more general version which has already been applied in the theory of quantum graphs (see, e.g., [44]).…”