Spectral Theory of Infinite Quantum Graphs

Exner, Pavel; Kostenko, Aleksey; Malamud, M. M.; Neidhardt, Hagen

doi:10.1007/s00023-018-0728-9

Cited by 45 publications

(72 citation statements)

References 100 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Choosing an increasing sequence {G n } ⊆ K G such that every G 0 ∈ K G is eventually contained in G n and applying the same limit argument as before, we arrive at the second inequality in (4.6). Combining (4.5) with Corollary 3.5, we obtain Theorem 4.18 from [26].…”

Section: Connections With Discrete Isoperimetric Constantsmentioning

confidence: 60%

See 1 more Smart Citation

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

Self Cite

View full text Add to dashboard Cite

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

show abstract

Section: Connections With Discrete Isoperimetric Constantsmentioning

confidence: 60%

“…In this case the operator H 0 is not necessarily essentially self-adjoint (that is, its closure may have nonzero deficiency indices) and finding self-adjointness criteria is a challenging open problem. The next results were proved recently in [26]. Define the weight function m :…”

Section: Now Consider the Maximal Operator On G Defined Bymentioning

confidence: 96%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

Self Cite

View full text Add to dashboard Cite

show abstract

“…where the quantity on the right is supported within supp(η ) =: S, and can therefore be estimated in terms of η ∞ , η ∞ , sup(F )χ S , and ψ H 1 (S) . With a little algebraic juggling, we can rewrite (17) so that the derivatives ψ and η do not appear:…”

Section: Landscape Upper Bounds On Tunneling Regions Using Agmon's Mmentioning

confidence: 99%

Localization and landscape functions on quantum graphs

Harrell

Maltsev

2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We discuss explicit landscape functions for quantum graphs. By a "landscape function" Υ(x) we mean a function that controls the localization properties of normalized eigenfunctions ψ(x) through a pointwise inequality of the formThe ideal Υ is a function that a) responds to the potential energy V (x) and to the structure of the graph in some formulaic way; b) is small in examples where eigenfunctions are suppressed by the tunneling effect; and c) relatively large in regions where eigenfunctions may -or may not -be concentrated, as observed in specific examples. It turns out that the connectedness of a graph can present a barrier to the existence of universal landscape functions in the high-energy régime, as we show with simple examples. We therefore apply different methods in different régimes determined by the values of the potential energy V (x) and the eigenvalue parameter E.

show abstract

“…Спектральная теория квантовых графов с конечным или бесконечным числом рёбер активно развивается в последние два-три десятилетия (см. монографии [2,13], работы [3,4,7] и цитируемую в них литературу). Всюду в сообщении G = ( , )…”

Section: математикаunclassified

Quantum graphs with summable matrix potentials

Granovskyi

Malamud

Neidhardt

2019

Доклады Академии наук

View full text Add to dashboard Cite

Let G be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume also that the length at least of one of the edges is infinite. The main object of the paper is Hamiltonian Hα associated in L2(G; Cm) with matrix Sturm-Liouville’s expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying technique of boundary triplets and the corresponding Weyl functions we show that the singular continuous spectrum of the Hamiltonian Hα as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring the positive part of the Hamiltonian Hα to be purely absolutely continuous. Under an additional condition on the potential matrix the Bargmann type estimate for the number of the negative eigenvalues of the operator Hα is obtained. Also we find a formula for the scattering matrix of the pair {Hα, HD} where HD is the operator of the Dirichlet problem on the graph.

show abstract

Spectral Theory of Infinite Quantum Graphs

Cited by 45 publications

References 100 publications

Spectral estimates for infinite quantum graphs

Spectral estimates for infinite quantum graphs

Localization and landscape functions on quantum graphs

Quantum graphs with summable matrix potentials

Contact Info

Product

Resources

About