Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians

Dodziuk, Józef; Mathai, Varghese

doi:10.1090/conm/398/07484

Cited by 48 publications

(73 citation statements)

References 16 publications

(25 reference statements)

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“…Using the substitution t → u 2 t, we arrive at the expression log (s/u) 2 + Λ j = We can now employ bounds from Section 5 in order to determine the asymptotic behavior of (84) [15] easily extend to compute the spectrum of the Laplacian on DT B in terms of the dual lattice B * . The existence and uniqueness of the associated heat kernel on DT B follow from general results (see, e.g., [6], [7]), thus allowing one to extend the results of Section 3 above. Assume there is a one-parameter family DT B(u) of discrete tori parametrized by u ∈ Z such that B(u)/u → M as u → ∞, where M is a positive definite d × d matrix.…”

Section: The Epstein-hurwitz Zeta Functionmentioning

confidence: 72%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

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Abstract. By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz-type zeta functions.

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Section: The Epstein-hurwitz Zeta Functionmentioning

confidence: 72%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

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“…If we set m V (v) = 1 for each vertex v ∈ V and m A (e) = 1 for each edge e ∈ A, then the magnetic combinatorial Laplacian is expressed by (1.4) and is discussed in Sections 1-4. This magnetic Laplacian and corresponding Schrödinger operators are considered in [B13], [DM06], [LL93].…”

Section: Properties Of Fiber Operators and An Examplementioning

confidence: 99%

Magnetic Schrödinger operators on periodic discrete graphs

Korotyaev

Saburova

2017

Journal of Functional Analysis

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Abstract. We consider magnetic Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schrödinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schrödinger operators constructed in the paper.

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“…We assume that G has bounded vertex degree: there exists a constant N > 0 such that m(x) ≤ N, for all x ∈ V . (1) In what follows, x ∼ y indicates that there is an edge that connects x and y. We will also need a set of oriented edges [y, x] : x, y ∈ V and x ∼ y}.…”

Section: The Settingmentioning

confidence: 99%

“…For the case a ≡ 1 and w ≡ 1, the definition (4) is the same as in [1]. For the case σ ≡ 1, see [2,3].…”

Section: A Magnetic Schrödinger Operatormentioning

confidence: 99%

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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Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians

Cited by 48 publications

References 16 publications

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Magnetic Schrödinger operators on periodic discrete graphs

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

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