On the Spectral Gap for Laplacians on Metric Graphs

• Authors: P. Kurasov
• Languages of publication: EN

Abstracts

EN

We discuss lower and upper estimates for the spectral gap of the Laplace operator on a finite compact connected metric graph. It is shown that the best lower estimate is given by the spectral gap for the interval with the same total length as the original graph. An explicit upper estimate is given by generalizing Cheeger's approach developed originally for Riemannian manifolds.

Keywords

EN

03.65.Nk; 73.63.-b; 85.35.-p

Discipline

• 73.63.-b: Electronic transport in nanoscale materials and structures(see also 73.23.-b Electronic transport in mesoscopic systems)
• 03.65.Nk: Scattering theory
• 85.35.-p: Nanoelectronic devices

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2013
• Volume: 124
• Issue: 6
• Pages: 1060-1062

Dates

published

2013-12

Contributors

author

P. Kurasov

• Dept. of Mathematics, Stockholm Univ., 106 91, Stockholm, Sweden

References

• [1] M. Fiedler, Czechoslov. Math. J. 23, 298 (1973)
• [2] G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, AMS, Providence 2013
• [3] P. Kurasov, 'Quantum Graphs: Spectral Theory and Inverse Problems', in preparation
• [4] O. Post, Lecture Notes Math., Vol. 2039, Springer, Heidelberg 2012
• [5] G. Berkolaiko, P. Kuchment, Proc. Symp. Pure Math. 84, 117 (2012)
• [6] P. Exner, M. Jex, Phys. Lett. A 376, 713 (2012)
• [7] L. Friedlander, Ann. Inst. Fourier (Grenoble) 55, 199211 (2005)
• [8] P. Kurasov, G. Malenova, S. Naboko, J. Phys. A, Math. Theor. 46, 275309 (2013)
• [9] P. Kurasov, S. Naboko, accepted for publication in J. Spectral Theory
• [10] L. Euler, 'Solutio problematis ad geometriam situs pertinentis', Comment. Academiae Sci. I. Petropolitanae 8, 128 (1736)
• [11] C. Hierholzer, Chr. Wiener, 'Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren', Math. Ann. 6, 3032 (1873) (in German)
• [12] J. Cheeger, in: Problems in Analysis, Ed. R.C. Gunning, Princeton University Press, New Jersey 1970, p. 195

Identifiers

DOI

10.12693/APhysPolA.124.1060