On Properties of the Asymptotic Expansion of the Heat Trace for the N/D Problem

Dowker, J. S.; Kirsten, Klaus; Gilkey, Peter Β.

doi:10.1142/s0129167x01000927

Cited by 19 publications

(26 citation statements)

References 19 publications

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“…However, Seleey [40] has shown recently that such terms do not appear and there is classical asymptotic expansion in half-integer powers of t only. This seems to contradict the conclusions of [21], where it has been shown that such an expansion with locally computable coefficients does not exist. The term 'locally computable' is confusing though.…”

Section: Discussionmentioning

confidence: 60%

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Heat Kernel Asymptotics of Zaremba Boundary Value Problem

Avramidi

2004

Mathematical Physics, Analysis and Geometry

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The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with non-smooth (singular) boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the global parametrix of the heat equation is described in detail and the leading parametrix is computed explicitly. Some of the first non-trivial coefficients of the heat kernel asymptotic expansion are computed explicitly.

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Section: Discussionmentioning

confidence: 60%

“…The boundary operator is then discontinuous at the intersection of these parts. The boundary value problems of this type are called Zaremba problem in the literature [14,15] (see also [42,25,5,21,20]). …”

Section: Remarkmentioning

confidence: 99%

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

Avramidi

2004

Mathematical Physics, Analysis and Geometry

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“…By construction, S bEMI satisfies (4) with identical boundary conditions B = B , although we are being agnostic about the physical meaning of the boundary condition since we do not know what theory has an entanglement entropy given by the bEMI. Note that we refer to (52) and (54) as entanglement entropies, but keep in mind that the EMI and bEMI ansatzes can be extended to general Rényi entropies by replacing s 0 with s 0,n .…”

Section: Extensive Mutual Information Modelmentioning

confidence: 99%

Relating bulk to boundary entanglement

Berthiere

Witczak-Krempa

2019

Phys. Rev. B

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Quantum field theories have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in 2 + 1 dimensions, we show that the corner-induced logarithmic terms of free CFTs and certain Lifshitz theories are simply related to those that arise when the corner touches the boundary. We test our findings on the lattice, including a numerical implementation of Neumann boundary conditions. We also propose an ansatz, the boundary Extensive Mutual Information model, for a CFT with a boundary whose entanglement entropy is purely geometrical. This model shows the same bulk-boundary connection as Dirac fermions and certain supersymmetric CFTs that have a holographic dual. Finally, we discuss how our results can be generalized to all dimensions as well as to massive quantum field theories. Contents

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“…The task of finding eigenmodes ( E n , f n ), n = 1,2,…, of the Laplacian in the two‐dimensional (2D) and three‐dimensional (3D) domain Ω with mixed Dirichlet

{(|, f_{n} (r))}_{\partial {normalΩ}_{D}} = 0

and Neumann

{(|, boldn \nabla f_{n} (r))}_{\partial {normalΩ}_{N}} = 0

boundary conditions on its confining surface (for 3D) or line (for 2D) ∂ Ω= ∂ Ω D ∪ ∂ Ω N (where r = ( x , y ) for two dimensions or r = ( x , y , z ) for three dimensions, and n is a unit normal vector to ∂ Ω) is commonly referred to as Zaremba problem ; it is a known mathematical problem science. Apart from the purely mathematical interest, an analysis of such solutions is of a large practical significance as they describe miscellaneous physical systems.…”

Section: Introductionmentioning

confidence: 99%

A quantum waveguide with Aharonov–Bohm magnetic field

Najar

Raissi

2015

Math Methods in App Sciences

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In a previous study [5] we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. In the present work we study the effect of the presence of a magnetic field of Aharonov-Bohm type on this system. Precisely we prove that in the presence of such field there is some critical values of a 0 > 0, for which we have absence of the discrete spectrum for 0 < a d < a 0 . We give a sufficient condition for the existence of discrete eigenvalues.

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On Properties of the Asymptotic Expansion of the Heat Trace for the N/D Problem

Cited by 19 publications

References 19 publications

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

Relating bulk to boundary entanglement

A quantum waveguide with Aharonov–Bohm magnetic field

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