Propagation of polyhomogeneity, diffraction, and scattering on product cones

IF 1 3区数学 Q1 MATHEMATICS

Journal of Spectral Theory Pub Date : 2020-04-15 DOI:10.4171/jst/404

Mengxuan Yang

引用次数: 4

Abstract

We consider diffraction of waves on a product cone. We first show that diffractive waves enjoy a one-step polyhomogeneous asymptotic expansion, which is an improvement of Cheeger-Taylor's classical result of half-step polyhomogeneity of diffractive waves in [CT82a], [CT82b]. We also conclude that on product cones, the scattering matrix is the diffraction coefficient, which is the principal symbol of the diffractive half wave kernel, for strictly diffractively related points on the cross section. This generalize the result of Ford, Hassell and Hillairet in 2-dimensional flat cone settings [FHH18]. In the last section, we also give a radiation field interpretation of the relationship between the scattering matrix and the diffraction coefficient.

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多均匀性、衍射和散射在产品锥上的传播

我们考虑波在乘积锥上的衍射。我们首先证明了衍射波具有一步多均匀渐近展开，这是对[CT82a]、[CT82b]中Cheeger-Taylor关于衍射波半步多均匀性的经典结果的改进。我们还得出结论，在乘积锥上，散射矩阵是横截面上严格衍射相关点的衍射系数，它是衍射半波核的主要符号。这推广了Ford、Hassell和Hillairet在二维平锥设置[FH18]中的结果。在最后一节中，我们还对散射矩阵和衍射系数之间的关系进行了辐射场解释。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Spectral Theory MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.00

自引率

0.00%

发文量

期刊介绍： The Journal of Spectral Theory is devoted to the publication of research articles that focus on spectral theory and its many areas of application. Articles of all lengths including surveys of parts of the subject are very welcome. The following list includes several aspects of spectral theory and also fields which feature substantial applications of (or to) spectral theory. Schrödinger operators, scattering theory and resonances; eigenvalues: perturbation theory, asymptotics and inequalities; quantum graphs, graph Laplacians; pseudo-differential operators and semi-classical analysis; random matrix theory; the Anderson model and other random media; non-self-adjoint matrices and operators, including Toeplitz operators; spectral geometry, including manifolds and automorphic forms; linear and nonlinear differential operators, especially those arising in geometry and physics; orthogonal polynomials; inverse problems.