Singular spectral curves in finite gap integration

A singular integration by parts formula for the exponential Euclidean QFT on the plane

We give a novel characterization of the Euclidean quantum field theory with exponential interaction $ν$ on $\mathbb{R}^2$ through a renormalized integration by parts (IbP) formula, or otherwise said via an Euclidean Dyson-Schwinger equation for expected values of observables. In order to obtain the well-posedness of the singular IbP problem, we import some ideas used to analyse singular SPDEs and we require the measure to "look like" the Gaussian free field (GFF) in the sense that a suitable Wasserstein distance from the GFF is finite. This guarantees the existence of a nice coupling with the …

Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(ω+ x)^ρ \, \mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite par…

作为积分参数分布的主值意义下的积分

Integral in the sense of principal value as a distribution over parameters of integration

在奇异函数或奇异函数乘积的主值意义下的积分在积分参数的某些取值范围内可以表现为奇异函数。在这种情况下，如果随后需要对参数进行积分，则会出现关于将初始积分解释为积分参数上的分布的问题。通过在量子场论中的实际应用，提出了一种解决这一问题的方法。

具有奇异谱曲线的二维有限能隙势Schrodinger和Dirac算子

On two-dimensional finite-gap potential Schrodinger and Dirac operators with singular spectral curves

我们描述了二维势场薛定谔和狄拉克算符，它们在一个能级上是有限能隙的，具有奇异谱曲线。看起来奇点可能相当复杂。这种狄拉克算子表现为浸入三维空间的环面的谱曲线。

Symbolic calculus for singular curve operators

We define a generalization of the T\''oplitz quantization, suitable for operators whose T\''oplitz symbols are singular. We then show that singular curve operators in Topological Quantum Fields Theory (TQFT) are precisely generalized T\''oplitz operators of this kind and we compute for some of them, and conjecture for the others, their main symbol, determined by the associated classical trace function.

Singular Points and Singular Curves in von Kármán Elastic Surfaces

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von K{á}rm{á}n plates and generalize the classical von K{á}rm{á}n equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possib…