Quantum graphs on radially symmetric antitrees (Q2247196)

In the present study the authors mainly focused their attention on antitrees from the perspective of quantum graphs and discussed a detailed spectral analysis of the Kirchhoff Laplacian on radially symmetric antitrees. Antitrees come into sight in the investigation of discrete Laplacians and attracted a noteworthy attention especially after the work of \textit{K.-T. Sturm} [J. Reine Angew. Math. 456, 173--196 (1994; Zbl 0806.53041)]. Also, Kostenko and Nicolussi considered the approach intorudced by [\textit{V. A. Mikhailets}, Funct. Anal. Appl. 30, No. 2, 144--146 (1996; Zbl 0874.34069); translation from Funkts. Anal. Prilozh. 30, No. 2, 90--93 (1996); \textit{B. Muckenhoupt}, Stud. Math. 44, 31--38 (1972; Zbl 0236.26015)] for radially symmetric trees and used some ideas from [\textit{J. Breuer} and \textit{N. Levi}, Ann. Henri Poincaré 21, No. 2, 499--537 (2020; Zbl 1432.05061)], where discrete Laplacians on radially symmetric ``weighted'' graphs have been analyzed. To summarize in general terms, in this paper, after recalling some necessary definitions and presenting an hypothesis, the authors studied characterization of self-adjointness and a complete description of self-adjoint extensions, spectral gap estimates and spectral types (discrete, singular and absolutely continuous spectrum). Next, they demonstrated their main results by considering two special classes of antitrees: (i) antitrees with exponentially increasing sphere numbers and (ii) antitrees with polynomially increasing sphere numbers.