TY - JOUR
T1 - Differences Between Robin and Neumann Eigenvalues on Metric Graphs
AU - Band, Ram
AU - Schanz, Holger
AU - Sofer, Gilad
N1 - Publisher Copyright:
© 2023, Springer Nature Switzerland AG.
PY - 2023
Y1 - 2023
N2 - We consider the Laplacian on a metric graph, equipped with Robin (δ -type) vertex condition at some of the graph vertices and Neumann–Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann–Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin–Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.
AB - We consider the Laplacian on a metric graph, equipped with Robin (δ -type) vertex condition at some of the graph vertices and Neumann–Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann–Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin–Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.
UR - http://www.scopus.com/inward/record.url?scp=85180217830&partnerID=8YFLogxK
U2 - 10.1007/s00023-023-01401-2
DO - 10.1007/s00023-023-01401-2
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AN - SCOPUS:85180217830
SN - 1424-0637
JO - Annales Henri Poincare
JF - Annales Henri Poincare
ER -