The number of eigenstates: counting function and heat kernel

Abstract

The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.

17 pages, 1 figure. v2: Equivalent forms of Eqs. (4.8) and (9.2) are added

References in corpus (26)

• Heat kernel expansion: user's manual
• Heat-kernel coefficients of the Laplace operator on the D-dimensional ball
• Effective Action of Vacuum: Semiclassical Approach
• Casimir energies for massive fields in the bag
• Casimir energy in the MIT bag model
• Casimir interaction of two plates inside a cylinder
• Heat Kernel Approach in Quantum Field Theory
• Nonsmoothness of the boundary and the relevant heat kernel coefficients
• On the vacuum energy of a spherical plasma shell
• Ultraviolet Sensitivity in Higher Dimensions
• Casimir effect in rugby-ball type flux compactifications
• UV Sensitivity in Supersymmetric Large Extra Dimensions: The Ricci-flat Case
• The $a_{3/2}$ heat kernel coefficient for oblique boundary conditions
• Heat kernel and number theory on NC-torus
• Quantum Effective Action in Spacetimes with Branes and Boundaries
• Can one hear the shape of the Universe?
• Chiral anomaly for local boundary conditions
• Self-force of a point charge in the space-time of a symmetric wormhole
• Worldline approach to quantum field theories on flat manifolds with boundaries
• Derivative expansion of the heat kernel in curved space
• Integral Equations for Heat Kernel in Compound Media
• Construction of the effective action in nonanticommutative supersymmetric field theories
• Nilpotent noncommutativity and renormalization
• Volume stabilization in a warped flux compactification model
• Conformal Enhancement of Holographic Scaling in Black Hole Thermodynamics: A Near-Horizon Heat-Kernel Framework
• Vacuum Structure and Global Strings with Conical Singularities

Cited by in corpus (30)

• The local potential approximation in quantum gravity
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• Exact solution of inverse-square-root potential $V\left(r\right)=-\frac{\alpha }{\sqrt{r}}$
• Scattering theory without large-distance asymptotics
• Scattering state and bound state of scalar field in Schwarzschild spacetime: Exact solution
• Scattering theory without large-distance asymptotics in arbitrary dimensions
• Scalar scattering in Schwarzschild spacetime: Integral equation method
• Heat-kernel approach for scattering
• Duality family of scalar field
• An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions
• Calculating eigenvalues of many-body systems from partition functions
• Semiclassical $p$-branes in hyperbolic space
• Scalar field in Reissner-Nordström spacetime: Bound state and scattering state
• Origin of quantum shape effect
• Bose-Einstein condensation temperature of finite systems
• Can one hear the density of a drum? Weyl's law for inhomogeneous media
• Energy spectrum of interacting gas: cluster expansion method
• Spectral properties of size-invariant shape transformation
• Bose-Einstein condensation in two-dimensional traps
• A 1+5-dimensional gravitational-wave solution: curvature singularity and spacetime singularity
• Probability Thermodynamics and Probability Quantum Field
• Covariant perturbation expansion of off-diagonal heat kernel
• Scattering approach for calculating one-loop effective action and vacuum energy
• Hear $\pi$ from Quantum Galperin Billiards
• Virial coefficients expressed by heat kernel coefficients
• Geodesic dual spacetime
• Heat kernel approach for confined quantum gas
• Renormalization for singular-potential scattering
• Statistical problem of ideal gas in general 2-dimensional regions
• Scattering theory without large-distance asymptotics: scattering boundary condition