相关论文: Regularized traces and Taylor expansions for the h…
In this work, a higher regularized trace formula has been found for a regular Sturm-Liouville differential operator with operator coefficient.
We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$,…
The purpose of this paper is to give some refined results about the distribution of resonances in potential scattering. We use techniques and results from one and several complex variables, including properties of functions of completely…
We use the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.
We present a brief overview of several approaches for calculating the local asymptotic expansion of the heat kernel for Laplace-type operators. The different methods developed in the papers of both authors some time ago are described in…
Let $m\in\mathbb N$, $P(D):=\sum_{|\alpha|=2m}(-1)^m a_\alpha D^\alpha$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the…
We consider the Schr\"odinger type operator ${\mathcal A}=(1+|x|^{\alpha})\Delta-|x|^{\beta}$, for $\alpha\in [0,2]$ and $\beta\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in…
We compute the full asymptotic expansion of the heat kernel Trace$(\exp(-tD^2))$ where $D$ is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The…
In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…
Using the technique of labeled operators, compact explicit expressions are given for all traced heat kernel coefficients containing zero, two, four and six covariant derivatives, and for diagonal coefficients with zero, two and four…
We obtain new results about the high-energy distribution of resonances for the one-dimensional Schr\"odinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular…
By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from…
The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients…
The covariant technique for calculating the heat kernel asymptotic expansion for an elliptic differential second order operator is generalized to manifolds with boundary. The first boundary coefficients of the asymptotic expansion which are…
We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in…
We study the heat kernel for an operator of Laplace type with a $\delta$-function potential concentrated on a closed surface. We derive the general form of the small $t$ asymptotics and calculate explicitly several first heat kernel…
In this paper we provide new asymptotic estimates of the Floquet exponents of Schr\"odinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.
Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several…
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…
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.