Essential Self-adjointness of Magnetic Schrödinger Operators on Locally Finite Graphs

Milatovic, Ognjen

doi:10.1007/s00020-011-1882-3

Cited by 42 publications

(50 citation statements)

References 36 publications

(58 reference statements)

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“…While the first reference does not allow magnetic fields and negative potentials, the second one assumes a uniformly bounded vertex degree, a condition that we will avoid by using the concept of intrinsic metrics. The proof works analogously to [34]. We refer also to [36] for results in this direction.…”

Section: Remark 217 (A) Theorem 216 Is a Generalization Of [23 Cormentioning

confidence: 90%

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

Güneysu

Keller

Schmidt

2015

Probab. Theory Relat. Fields

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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

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Section: Remark 217 (A) Theorem 216 Is a Generalization Of [23 Cormentioning

confidence: 90%

“…and [34,Theorem 1.5]. While the first reference does not allow magnetic fields and negative potentials, the second one assumes a uniformly bounded vertex degree, a condition that we will avoid by using the concept of intrinsic metrics.…”

Section: Remark 217 (A) Theorem 216 Is a Generalization Of [23 Cormentioning

confidence: 98%

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

Güneysu

Keller

Schmidt

2015

Probab. Theory Relat. Fields

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“…It is easy to see that for every k ∈ R, there exists a function u ∈ C c (V ) such that the inequality (10) is not satisfied. Thus, the operator H is not semi-bounded from below, and we cannot use [11,Theorem 1.3]. Turning to hypotheses of Theorem 1, note that W satisfies (7) with q(n) = n 2 .…”

Section: Remarkmentioning

confidence: 96%

“…Thanks to assumption (10), the proof of [11,Theorem 1.3] reduced to showing that if u ∈ Dom(H max ), with H max as in Section 3, and (H + λ)u = 0 with sufficiently large λ > 0, then u = 0. To this end, a sequence of cut-off functions was constructed and a ''summation by parts'' method was used.…”

Section: Remarkmentioning

confidence: 98%

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A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

Milatovic

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn the context of a weighted graph with vertex set V and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator ∆ σ + W , where ∆ σ is the magnetic Laplacian and W : V → R is a function satisfying W (x) ≥ −q(x) for all x ∈ V , with q: V → [1, ∞). The condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M.A. Shubin in the setting of non-compact Riemannian manifolds.

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Intrinsic Metrics on Graphs: A Survey

Keller

2015

Springer Proceedings in Mathematics &Amp; Statistics

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A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.

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Essential Self-adjointness of Magnetic Schrödinger Operators on Locally Finite Graphs

Abstract: We give sufficient conditions for essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Two of the main results of the present paper generalize recent results of Torki-Hamza. Mathematics Subject Classification (2000). Primary 35J10, 47B25; Secondary 05C63.

Cited by 42 publications

References 36 publications

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

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

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

Intrinsic Metrics on Graphs: A Survey

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