Narrow Results

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph-theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich’s formula.

In the setting of finite-dimensional $\mathrm{\text{RCD}}(K,N)$ spaces, we characterize the $p$-Sobolev spaces for $p\in (1,\infty )$ and the space of functions of bounded variation in terms of the short-time behavior of the heat flow. Moreover, we prove that Cheeger $p$-energies and total variations can be computed as limits of nonlocal functionals involving the heat kernel.

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as $d$-dimensional balls (with $d=1,2,3$), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator $L$ directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function $\zeta (s)=\mathrm{\text{Tr}}{L}^{-s}$ and the heat trace $\mathrm{\Theta }(t)=\mathrm{\text{Tr}}exp(-tL)$. The kernel $U(t;x,x^{\prime })$ of the heat semigroup $exp(-tL)$, called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as $\mathrm{\text{Tr}}f(tL)$, that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as $\mathrm{\text{Tr}}exp(-t{L}_{+})exp(-s{L}_{-})$, that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

We describe the heat kernel asymptotics for roots of a Laplace type operator Δ on a closed manifold. A previously known relation between the Wodzicki residue of Δ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form ${\overline{\square }}_{\phi }=\overline{D}{\overline{D}}^{\ast }$, where $\overline{D}f=\bar{\partial }f+{\phi }_{\bar{z}}f$, the function $\phi$ is smooth and subharmonic on $ℂ$, and ${\overline{D}}^{\ast }$ is the formal adjoint of $\overline{D}$. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in $ℝ\times ℂ$, J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

The spectrum of the fermionic operators depending on external fields is an important object in quantum field theory. In this paper we prove, using transition to the alternative basis for the γ-matrices, that this spectrum does not depend on the sign of the fermion mass, up to a constant factor. This assumption has been extensively used, but usually without proof. As an illustration, we calculated the coincidence limit of the coefficient a₂(x,x′) on the general background of metric, vector and axial vector fields.

We compute a Chern–Simons term induced by the fermions on noncommutative torus interacting with two U(1) gauge fields. For rational noncommutativity θ∝P/Q we find a new mixed term in the action which involves only those fields which are (2π)/Q periodic, like the fields in a crystal with Q² nodes.

We show that there is no superradiance for the Dirac field in the rotating BTZ black hole if the field vanishes at infinity. Then we outline the calculation of the expression for the renormalized energy–momentum tensor, the effective action as well as the heat kernel for the Dirac field for the BTZ black hole. Finally, we point out how to construct the Hartle–Hawking–Israel state for the real scalar field in the non-rotating BTZ black hole in two and three dimensions.

In this paper, based on the heat kernel technique, we calculate equations of state and thermodynamic quantities for ideal quantum gases in confined space with external potential. Concretely, we provide expressions for equations of state and thermodynamic quantities by means of heat kernel coefficients for ideal quantum gases. Especially, using an analytic continuation treatment, we discuss the application of the heat kernel technique to Fermi gases in which the expansion diverges when the fugacity $z>1$. In order to calculate the modification of heat kernel coefficients caused by external potentials, we suggest an approach for calculating the expansion of the global heat kernel of the operator $-\mathrm{\Delta }+U(x)$ based on an approximate method of the calculation of spectrum in quantum mechanics. We discuss the properties of quantum gases under the condition of weak and complete degeneration, respectively. Moreover, we give an expansion of the one-loop effective action in D-dimensional space.

For positive elliptic differential operators Δ, the asymptotic expansion of the heat trace tr(e^-tΔ) and its related zeta function ζ(s, Δ) = tr(Δ^-s) have numerous applications in geometry and physics. This article discusses the general nature of the boundary conditions that must be considered when there is a singular stratum, and presents three examples in which a choice of boundary conditions at the singularity must be made. The first example concerns the signature operator on a manifold with a singular stratum of conic type. The second concerns the "Zaremba problem" for a nonsingular manifold with smooth boundary, posing Dirichlet conditions on part of the boundary and Neumann conditions on the complement; the intersection of these two regions can be viewed as a singular stratum of conic type, and a boundary condition must be imposed along this stratum. The third example is a one-dimensional manifold where the operator at one end has a singularity like that in conic problems, and the choice of boundary conditions affects not just the residues at the poles of the zeta function, but also the very location of the poles

We generalize the Endo formula¹ originally developed for the computation of the heat kernel asymptotic expansion for nonminimal operators in commutative gauge theories to the noncommutative case. In this way, the first three nonzero heat trace coefficients of the nonminimal U(N) gauge field kinetic operator on the Moyal plane taken in an arbitrary background are calculated. We show that the nonplanar part of the heat trace asymptotics is determined by U(1) sector of the gauge model. The nonplanar or mixed heat kernel coefficients are shown to be gauge-fixing dependent in any dimension of space–time. In the case of the degenerate deformation parameter the lowest mixed coefficients in the heat expansion produce nonlocal gauge-fixing dependent singularities of the one-loop effective action that destroy the renormalizability of the U(N) model at one-loop level. Such phenomenon was observed at first in Ref. 2 for spacelike noncommutative ϕ⁴ scalar and U(1) gauge theories. The twisted-gauge transformation approach is discussed.

We review the general heat kernel method for the Dirac spinor field as an elementary example in any arbitrary background. We, then compute the first three Seeley–DeWitt coefficients for the massless free spin-3/2 Rarita–Schwinger field without imposing any limitations on the background geometry.

In chiral Einstein–Cartan gravity, a new gauge fixing procedure is implemented recently, leading to a very economical perturbation expansion of the action. Using this formulation and the relevant gauge fixing, we develop the ghost Lagrangian on an arbitrary Einstein background using the Becchi-Rouet-Stora-Tyutin (BRST) formalism. The novelty is the appearance of a new term quadratic in the tetrad field. We next compute the heat kernel coefficients and understand the divergences arising in the gravitational one-loop effective action. In our computation, the arising heat kernel coefficients depend only on the self-dual part of the Weyl curvature. We make a comparison between our results and what has been obtained for metric general relativity.

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

Diffusions on the pinned loop and pathgroup of a general (non-compact) Lie group are constructed. The laws of the endpoint processes satisfy heat equations related to second order differential operators associated to subspaces of the Cameron–Martin space.

An elementary proof is given for the positivity of the heat kernel associated with the sub-Laplacian on Heisenberg group. Starting from an integral representation of the heat kernel, the positivity is shown by choosing a simple, explicit path of integration. The positivity of the heat kernel associated with the Grušin operator then follows.

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker’s limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.