Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

Mathematische Nachrichten

Münch

Pogorzelski

2016

Abstract. We study graphs whose vertex degree tends and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness.

“…This yields the bound in Theorem . Furthermore, from [, Theorem 5.3] it follows that for all

ε > 0

there is

C_{ε} > 0

such that for all normalized functions φ with finite support

\begin{matrix} true \\ left (1 - ε) (() 1 - \sqrt{1 - α_{\infty}^{2}}) ⟨ deg φ, φ ⟩ - C_{ε} \leq ⟨ Δ φ, φ ⟩ \\ left \leq (1 + ε) (() 1 + \sqrt{1 - α_{\infty}^{2}}) ⟨ deg φ, φ ⟩ + C_{ε} . \end{matrix}

Section: Isoperimetric Constantsmentioning

confidence: 99%

Section: Isoperimetric Constantsmentioning

confidence: 99%

“…Remark Let us briefly comment on how the class of graphs with

r_{\infty} < 1

r_{\infty} < 1

and planar graphs, obviously, neither includes the other.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometry and spectrum of rapidly branching graphs

Mathematische Nachrichten

Münch

Pogorzelski

2016

“…Magnetic Schrödinger operators on graphs have been intensively studied in recent years. The topics of research range from essential self adjointness [Mi1,Mi2,CTT,G] over Feynman-Kac-Itô formulas [GKS, GMT] to spectral considerations [DM, GT, LLPP, LMP].…”

Section: Introductionmentioning

confidence: 99%

Magnetic-Sparseness and Schrödinger Operators on Graphs

Bonnefont

Golenia

et al. 2020

Ann. Henri Poincaré

Self Cite

“…In the last years an extensive amount of research for these operators has been carried out into various directions. Let us only mention here that basic spectral properties and Kato's inequality have been proven in [7], for a Hardy inequality see [12], for approximation results of spectral invariants see [32,33], and for weak Bloch theory see [20]. Recently there has been a strong focus on the question of essential self-adjointness of magnetic Schrödinger operators [4,12,[34][35][36]45].…”

Section: Introductionmentioning

confidence: 99%

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

Güneysu

Probab. Theory Relat. Fields

Schmidt

2015

In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödi-nger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials. Mathematics Subject Classification