Mathematical aspects of vacuum energy on quantum graphs

Berkolaiko, Gregory; Harrison, J. M.; Wilson, Justin H.

doi:10.1088/1751-8113/42/2/025204

Cited by 18 publications

(37 citation statements)

References 49 publications

(107 reference statements)

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“…This agrees with the result of [7] (also see [3,9]) where the vacuum energy of an equilateral star graph was calculated. As the Casimir force is proportional to the derivative of E c with respect to L, this example was used to demonstrate that the Casimir force changes from attractive to repulsive depending on the number of edges.…”

Section: Figuresupporting

confidence: 91%

Relating zeta functions of discrete and quantum graphs

Harrison

Weyand

2017

Lett Math Phys

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We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation between the spectrum of the Laplacian on a discrete graph and that of the Laplacian on an equilateral metric graph. As a by-product, we determine how the multiplicity of eigenvalues of the quantum graph, that are also in the spectrum of the graph with Dirichlet conditions at the vertices, depends on the graph geometry. Finally we apply the result to calculate the vacuum energy and spectral determinant of a complete bipartite graph and compare our results with those for a star graph, a graph in which all vertices are connected to a central vertex by a single edge.

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Section: Figuresupporting

confidence: 91%

Relating zeta functions of discrete and quantum graphs

Harrison

Weyand

2017

Lett Math Phys

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“…According to (3.3) the cases that give rise to r,t ∈ (0, 1) when n = 5 are (1, 0), (4, 5), (2, 0), (3, 5), (2, 1) and (3,4). Each case is discussed in the sequel.…”

Section: Ordermentioning

confidence: 99%

On Equi-Transmitting Matrices

Kurasov

Ogik

2016

Reports on Mathematical Physics

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“…Some results of that research have been published in [9,22,23], to which we refer for details. We are happy to acknowledge strong interactions with, and assistance from, the Texas A&M quantum graph research group…”

Section: Overview and Acknowledgmentsmentioning

confidence: 99%

“…When the edge lengths are not equal, exact spectral calculations are more difficult. However, the cylinder kernel can still be constructed exactly by an appropriate generalization of the method of images [43,9]. One starts with the "free" cylinder kernel on the real line,…”

Section: 2mentioning

confidence: 99%

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Vacuum energy and closed orbits in quantum graphs

Fulling

Wilson²

2008

Analysis on Graphs and Its Applications

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Abstract. The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos-Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scale-invariant boundary conditions the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general (frequency-dependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be "indexed" a posteriori by truly periodic orbits. For the scale-invariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula. Overview and acknowledgmentsVacuum energy [15,14,38,37,13] is a topic in physics (quantum field theory) that has turned out to have some resonance in mathematics. It provides an application of, and a new window on, the spectral theory of second-order differential operators. In particular, there are connections with the lore surrounding the relations between the operator's spectrum and the trajectories of an associated classical-mechanical system. Insofar as energy, pressure, etc. can be localized in space, the study of vacuum energy promotes attention to localized spectral information, a somewhat neglected topic. The present article is complementary to [21]; both are directed primarily to a mathematical audience.Much of the article is based on the undergraduate research [43] of J. Wilson under the direction of S. Fulling and G. Berkolaiko. Some results of that research have been published in [9,22,23], to which we refer for details. We are happy to acknowledge strong interactions with, and assistance from, the Texas A&M quantum graph research group

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Mathematical aspects of vacuum energy on quantum graphs

Cited by 18 publications

References 49 publications

Relating zeta functions of discrete and quantum graphs

Relating zeta functions of discrete and quantum graphs

On Equi-Transmitting Matrices

Vacuum energy and closed orbits in quantum graphs

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