Spectral functions in mathematics and physics

Abstract

Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed. Examples comprise heat-kernels, determinants and spectral sums needed for the analysis of Casimir energies. First, we summarize that a convenient way of handling them is to use the associated zeta function. A way to determine all its needed properties is derived. Using the connection with the mentioned spectral functions, we provide: i.) a method for the calculation of heat-kernel coefficients of Laplace-like operators on Riemannian manifolds with smooth boundaries and ii.) an analysis of vacuum energies in the presence of spherically symmetric boundaries and external background potentials.

51 pages, Invited talk at 2nd La Plata Meeting on Trends in Theoretical Physics, Buenos Aires, Argentina, November-December 1998, Published in AIP Conference Proceedings 484, page 106, Edited by H. Falomir, R.E. Gamboa Saravi and F.A. Schaposnik

References in corpus (5)

• On the ground state energy for a penetrable sphere and for a dielectric ball
• The ground state energy of a spinor field in the background of a finite radius flux tube
• Dirac fields in the background of a magnetic flux string and spectral boundary conditions
• Smeared heat-kernel coefficients on the ball and generalized cone
• Dependence of the vacuum energy on spherically symmetric background fields

Cited by in corpus (52)

• The generalized Abel-Plana formula with applications to Bessel functions and Casimir effect
• Surface Casimir densities and induced cosmological constant on parallel branes in AdS
• Fermionic Casimir effect for parallel plates in the presence of compact dimensions with applications to nanotubes
• Selected Topics in the Large Quantum Number Expansion
• Induced fermionic current in toroidally compactified spacetimes with applications to cylindrical and toroidal nanotubes
• Fermionic Casimir densities in toroidally compactified spacetimes with applications to nanotubes
• Non-commutative heat kernel
• Casimir energy-momentum tensor for a brane in de Sitter spacetime
• Gravitational parity anomaly with and without boundaries
• Effective action for the Yukawa model in curved spacetime
• Multiple reflection expansion and heat kernel coefficients
• Strongly Interacting Fermions and Phases of the Casimir Effect
• Fermionic Casimir densities in anti-de Sitter spacetime
• Parity anomaly in four dimensions
• Casimir energy in the Fulling--Rindler vacuum
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Vacuum Polarization Energy for General Backgrounds in One Space Dimension
• Casimir densities for two concentric spherical shells in the global monopole spacetime
• Casimir densities for a spherical shell in the global monopole background
• Casimir energy-momentum tensor for the Robin Surfaces in de Sitter Spacetime
• Exact one-loop false vacuum decay rate
• Analytic thin wall false vacuum decay rate
• Gauged Yukawa model in curved spacetime
• Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension
• Casimir Pistons with General Boundary Conditions
• Zeta Determinant for Laplace Operators on Riemann Caps
• A Non-Singular One-Loop Wave Function of the Universe From a New Eigenvalue Asymptotics in Quantum Gravity
• Spectral Methods for Coupled Channels with a Mass Gap
• Heat kernel coefficients for massive gravity
• Vacuum polarization energy of the Shifman-Voloshin soliton
• Functional Determinants of Radial Operators in $Ad{S}_2$
• Small Mass Expansion of Functional Determinants on the Generalized Cone
• Volume stabilization in a warped flux compactification model
• Asymptotic Expansion of the Heat Kernel Trace of Laplacians with Polynomial Potentials
• Null energy condition violation: Tunnelling versus the Casimir effect
• Casimir force in the Gödel space-time and its possible induced cosmological inhomogeneity
• On the Hurwitz Zeta Function of Imaginary Second Argument
• Extremal metrics for spectral functions of Dirac operators in even and odd dimensions
• Casimir effect in a two dimensional signature changing spacetime
• Spontaneous magnetization of the vacuum and the strength of the magnetic field in the hot Universe
• Partition function for a singular background
• Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds
• Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring
• On the Casimir energy for a massive quantum scalar field and the Cosmological constant
• Casimir effect for a spherical shell in de Sitter spacetime with signature change
• One-loop analysis with nonlocal boundary conditions
• Heat Kernel Asymptotic Expansion on Unbounded Domains with Polynomially Confining Potentials
• Heat Kernel Asymptotics on Symmetric Spaces
• Spectral Geometry and the Semiclassical Approach
• Casimir effect for concentric spheres in de Sitter spacetime with signature change
• Non-Perturbative Aspects of Quantum Electrodynamics on Curved Space and Investigations in Matrix Gravity
• Casimir Effect for Moving Branes in Static dS${}_{4+1}$ Bulk