Eigenvalue estimates for the Laplacian on a metric tree
Author:
Jonathan Rohleder
Journal:
Proc. Amer. Math. Soc. 145 (2017), 2119-2129
MSC (2010):
Primary 34B45; Secondary 81Q10, 81Q35
DOI:
https://doi.org/10.1090/proc/13403
Published electronically:
October 27, 2016
MathSciNet review:
3611325
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e., the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.
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Additional Information
Jonathan Rohleder
Affiliation:
TU Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, Gebäude E, 21073 Hamburg, Germany
Email:
jonathan.rohleder@tuhh.de
DOI:
https://doi.org/10.1090/proc/13403
Received by editor(s):
February 17, 2016
Received by editor(s) in revised form:
July 4, 2016
Published electronically:
October 27, 2016
Additional Notes:
The author wishes to thank Gregory Berkolaiko for drawing his attention to the eigenvalue inequalities contained in [1]. Moreover, the author gratefully acknowledges financial support from the Austrian Science Fund (FWF), project P 25162-N26.
Communicated by:
Michael Hitrik
Article copyright:
© Copyright 2016
American Mathematical Society