Integral representations of closed operators as bi-Carleman operators with arbitrarily smooth kernels

Spectral and scattering theory for perturbations of the Carleman operator

We study spectral properties of the Carleman operator (the Hankel operator with kernel $h_{0}(t)=t^{-1}$) and, in particular, find an explicit formula for its resolvent. Then we consider perturbations of the Carleman operator $H_{0}$ by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as $t\to\infty$ and not too singular at t=0. Our goal is to develop scattering theory for the pair $H_{0}$, $H=H_{0} +V $ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $H$. We also prove that under general assumptions the singular continuou…

Kernels of Integral Equations Can Be Boundedly Infinitely Differentiable on $\mathbb{R}^2$

In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,μ)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations.

An extension of Mercer's theorem to unbounded operators

We give some extensions of Mercer's theorem to continuous Carleman kernels inducing unbounded integral operators.

Quasi-diagonalization of Hankel operators

We show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+}) $ can be quasi-diagonalized as $H= {\sf L}^* Σ{\sf L} $. Here ${\sf L}$ is the Laplace transform, $Σ$ is the operator of multiplication by a function (distribution) $σ(λ)$, $λ\in {\Bbb R}$. We find a scale of spaces of test functions where ${\sf L} $ acts as an isomorphism. Then ${\sf L}^*$ is an isomorphism of the corresponding spaces of distributions. We show that $h= {\sf L}^* σ$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $σ(λ)$ of Hankel…

近似Tsirelson贝尔-CHSH违反在量子场论中通过Carleman和Hankel算子

Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Carleman and Hankel Operators

本文研究了自由旋量场真空态中的贝尔-CHSH违反，通过构造具有空间状分离支持的测试函数，证明了其收敛到Tsirelson界2√2，并探讨了Carleman和Hankel算子的谱理论与贝尔-CHSH违反的关系。

Manifold embeddings by heat kernels of connection Laplacian

We show that any closed $n$-dimensional manifold $(M,g)$ can be embedded by a map constructed using the heat kernels of the connection Laplacian as well as a maps constructed using truncated heat kernel at a certain time $t$ from a $δ$-net $\{q_i\}_{i=1}^{N_0}$ via a rescaling trick. Both the time $t$ and $N_0$ are bounded in terms of the dimension, bounds on the Ricci curvature and its derivative, the injectivity radius, and the volume. Moreover, both maps can be made arbitrarily close to an isometry.