scholarly journals Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

2021 ◽  
Vol 11 (4) ◽  
pp. 1857-1894
Author(s):  
Marvin Plümer
Keyword(s):  
Metric Graphs ◽  
Eigenvalue Bounds

scholarly journals On the nonlinear Dirac equation on noncompact metric graphs

2021 ◽  
Vol 278 ◽  
pp. 326-357
Author(s):  
William Borrelli ◽  
Raffaele Carlone ◽  
Lorenzo Tentarelli
Keyword(s):  
Dirac Equation ◽  
Metric Graphs ◽  
Nonlinear Dirac Equation

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

scholarly journals Computable eigenvalue bounds for rank-k perturbations

2010 ◽  
Vol 432 (12) ◽  
pp. 3100-3116 ◽  
Author(s):  
Jan H. Brandts ◽  
Ricardo Reis da Silva
Keyword(s):  
Eigenvalue Bounds

scholarly journals Infinite reduction of divisors on metric graphs

2014 ◽  
Vol 35 ◽  
pp. 67-74 ◽  
Author(s):  
Spencer Backman
Keyword(s):  
Metric Graphs

Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

1996 ◽  
Vol 106 (3) ◽  
pp. 281-287 ◽  
Author(s):  
Mihir B Banerjee ◽  
R G Shandil ◽  
Balraj Singh Bandral
Keyword(s):  
Eigenvalue Bounds ◽  
Backward Wave ◽  
Sommerfeld Equation

scholarly journals Heat kernels on metric graphs and a trace formula

2007 ◽  
pp. 175-198 ◽  
Author(s):  
Vadim Kostrykin ◽  
Jürgen Potthoff ◽  
Robert Schrader
Keyword(s):  
Trace Formula ◽  
Heat Kernels ◽  
Metric Graphs

scholarly journals Eigenvalue bounds for the signless laplacian

2007 ◽  
pp. 11-27 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Peter Rowlinson ◽  
Slobodan Simic
Keyword(s):  
Previous Survey ◽  
Eigenvalue Bounds ◽  
Largest Eigenvalue ◽  
Signless Laplacian

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

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