An accurate asymptotic estimate for the trace of the heat kernel of a $\Gamma$-pseudo-differential operator

Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

Preface.- Adaptive Wavelet Computations for Inverses of Pseudo-Differential Operators.- Spectral Theory of Pseudo-Differential Operators on S1.- A Characterization of Compact Pseudo-Differential Operators on S1.- Melin Operator with Asymptotics on Manifolds with Corners.- The Iterative Structure of the Corner Calculus.- Elliptic Equations and Boundary Value Problems in Non-Smooth Domains.- Calculus of Pseudo-Differential Operators and a Local Index of Dirac Operators.- Lp Bounds for a Class of Fractional Powers of Subelliptic Operators.- The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator.- Local Exponential Estimates for h-Pseudo-Differential Operators with Operator-Valued Symbols.- Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane.- Fuchsian Mild Microfunctions with Fractional Order and their Applications to Hyperbolic Equations.- The Continuity of Solutions with respect to a Parameter to Symmetric Hyperbolic Systems.- Generalized Gevrey Ultradistributions and their Microlocal Analysis.- Weyl Rule and Pseudo-Differential Operators for Arbitrary Operators.- Time-Frequency Characterization of Stochastic Differential Equations.- Wigner Representation Associated with Linear Transformations of the Time-Frequency Plane.- Some Remarks on Localization Operators.

We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators to construct Feller semigroups. Our method works with “rough” symbols $${p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)}$$ only needs to be continuous. The main part of this work concerns the development of an asymptotic expansion formula for the composition of two pseudodifferential operators with rough negative definite symbols. This presents an improvement over other symbolic calculi that typically require the symbols to be smooth. As an application we show how to adapt existing techniques to construct and approximate Feller semigroups to the case of rough symbols.

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\) pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.

We consider the heat kernel for higher-derivative and nonlocal operators in $d$-dimensional Euclidean space-time and its asymptotic behavior. As a building block for operators of such type, we consider the heat kernel of the minimal operator - generic power of the Laplacian - and show that it is given by the expression essentially different from the conventional exponential Wentzel-Kramers-Brillouin (WKB) ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright $\varPsi$-functions and Fox $H$-functions. The structure of its essential singularity in the proper time parameter is different from that of the usual exponential ansatz, which invalidated previous attempts to directly generalize the Schwinger-DeWitt heat kernel technique to higher-derivative operators. In particular, contrary to the conventional exponential decay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivative operators. We give several integral representations for the generalized exponential function, find its asymptotics and semiclassical expansion, which turns out to be essentially different for local operators and nonlocal operators of noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.

We compute the third order term in a generalization of the Strong Szegö Limit Theorem for a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold of an arbitrary dimension. In Guillemin and Okikiolu (Guillemin, V., Okikiolu, K. ([1997a]). Spectral asymptotics of Toeplitz operators on Zoll manifolds. J. Funct. Anal. 146:496–516), the second order term was computed by Guillemin and Okikiolu. In the present article, an important role is played by a certain combinatorial identity which we call the generalized Hunt–Dyson formula (Gioev, D. ([2002b]). Generalized Hunt–Dyson formula and Bohnenblust–Spitzer theorem. Int. Math. Res. Not. 2002(32):1703–1722). This identity is a different form of the renowned Bohnenblust–Spitzer combinatorial theorem which is related to the maximum of a random walk with i.i.d. steps on the real line. A corollary of our main result is a fourth order Szegö type asymptotics for a zeroth order PsDO on the unit circle, which in matrix terms gives a fourth order asymptotic formula for the determinant of the truncated sum of a Toeplitz matrix T 1 with the product of another Toeplitz matrix T 2 and a diagonal matrix D of the form . Here , n ones. †Dedicated to the memory of my grandparents, Valentina Nikolaevna Pavlova and Dimitri Aleksandrovich Golubentsev.

We consider non-local Schrödinger operators H=-L-V in L^2(mathbb {R}^d), d geqslant 1, where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.

Complex powers of hypoelliptic pseudodifferential operators

An essentially self-contained, rigorous proof of the Atiyah–Singer index theorem is given for the case of the twisted Dirac operator. Indeed, the stronger local index theorem, which implies the general case, is proven here by computing the supertrace of the steady (time-independent) asymptotic of the twisted spinorial heat kernel. The computation is carried out in the spirit of Patodi’s proof of the Gauss–Bonnet–Chern theorem. Gilkey’s approach involving invariant theory is not used, but rather a generalization of Mehler’s formula is derived and utilized along with some elementary properties of Clifford algebras. The use of Mehler’s formula was inspired by work of Getzler, but families of Clifford algebra-valued pseudodifferential operators, and limits and estimates of families of heat kernels, are avoided here. Asymptotic expansions of heat kernels for operators on Euclidean fields are of fundamental importance in the computation of quantum corrections to the effective action for classical fields. Thus, the general formula developed here for the asymptotics may also be helpful in this regard.

In this article we prove that the heat kernel attached to the non-Archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous $$C_{0}$$-transition function of some strong Markov processes $${\mathfrak {X}}$$ with state space $${\mathbb {Q}}_{p}^{n}.$$ We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a constant $$c\ge 0,$$ a continuous homomorphism $$l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}$$ and a non-negative, continuous quadratic form $$q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.$$

This research develops a new framework for modeling the dispersion of contaminants in non-Archimedean media using the Taibleson-Vladimirov Dαp-adic pseudo-differential operator. A p-adic differential equation is proposed and solved, modeling the dispersion of contaminants in a p-adic medium initially confined to a specific region. Additionally, it is analyzed how the p-adic heat kernel associated with the Taibleson-Vladimirov Dα operator serves as a powerful tool for understanding how contaminants disperse over time, making the heat kernel a cornerstone for mathematical models aimed at environmental applications in p-adic settings. Furthermore, through the heat equation associated with the Taibleson-Vladimirov Dα operator, the evolution of contaminant concentration over time is described. Finally, a stochastic fractional p-adic diffusion equation is formulated and solved, incorporating initial conditions, external sources, and random noise.

On the full calculus of pseudo-differential operators on boundary groupoids with polynomial growth

A coupling parameter is present in almost all the uniquely solvable boundary integral formulations of the Helmholtz equation in an exterior domain. Here we provide new insight into the well-known near optimal choice of this parameter by studying the asymptotic behaviour of the eigenvalues of the resulting boundary integral operators. By taking advantage of the recent results of Arnold and Wendland (1983, 1985) for the asymptotic convergence of collocation solution of strongly elliptic pseudo-differential operators we study the effect of numerical integration on the achievable accuracy of discrete collocation methods. We are then able to highlight an error in a recent paper by Kirkup (1992) which suggested that the coupling parameter should be inversely proportional to the number of collocation points.

The properties of pseudo-differential operators allow a unified and detailed analysis of numerical integration. We present error estimates exploiting the Rabinowitz-Richter estimates for three numerical integration techniques of boundary integral operators on an analytic boundary element containing the source point. We first study weakly singular integral operators which are pseudo-differential operators of order ≤ −1. We show that Duffy's triangular coordinates transform all these operators to regular integrals and provide asymptotic error estimates for corresponding Gaussian quadrature. Similar results can be obtained for plane polar coordinates in the parameter domain with scaled Gaussian quadrature in radial and Gaussian quadrature on the angular pieces of analyticity. We also find that Lyness extrapolation for all weakly singular boundary integral operators is asymptotically almost as efficient. For strongly singular integral operators defined by Cauchy singular and Hadamared finite part integrals which are pseudo-differential operators of orders ≥ 0 we analyze product integration with Kutt formulas in radial and Gaussian integration on the angular pieces of analyticity. Again, we give asymptotic error estimates.