Related papers: Toeplitz operators and the full asymptotic torsion…

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $\Phi_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii)…

This paper is a continuation of our previous work \cite{wang2024complex}. It mainly deals with entire operators $T$ with deficiency index 1 \emph{systematically} from the complex-geometric viewpoint proposed in \cite{wang2024complex}. We…

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-\Delta)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order…

We develop Berezin-Toeplitz quantization in a non-compact complex geometric setting. Let $(X,\Theta)$ be a Hermitian manifold, $(L,h^L)$ a positive holomorphic line bundle, and $(E,h^E)$ a holomorphic Hermitian vector bundle. Assuming that…

The main results of this paper are an asymptotic expansion in powers of $\hbar$ for the spectral measure $\mu_\hbar$ of a semi-classical Toeplitz operator, $Q_\hbar$, and an equivariant version of this result when $Q_\hbar$ admits an…

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[…

Bessel and modified Bessel functions of imaginary order $i\nu$ ($\nu >0$) are studied. Asymptotic expansions are derived as $\nu \to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with…

The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion…

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic…

We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense…

This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.

We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…

The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…

We consider the asymptotic expansion of the functional series \[S_{\mu,\gamma}(a;\lambda)=\sum_{n=1}^\infty \frac{n^\gamma e^{-\lambda n^2/a^2}}{(n^2+a^2)^\mu}\] for real values of the parameters $\gamma$, $\lambda>0$ and $\mu\geq0$ as…

Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large…

Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and…

We study the semiclassical Bochner-Schr\"odinger operator $H_{p}=\frac{1}{p^2}\Delta^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded…

We propose an asymptotic expansion formula for matrix integrals, including oscillatory terms (derivatives of theta-functions) to all orders. This formula is heuristically derived from the analogy between matrix integrals, and formal matrix…

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are…