Basics of the b-calculus

Microlocal analysis of operators with asymptotic translation- and dilation-invariances

On a suitable class of non-compact manifolds, we study (pseudo)differential operators which feature an asymptotic translation-invariance along one axis and an asymptotic dilation-invariance, or asymptotic homogeneity with respect to scaling, in all directions not parallel to that axis. Elliptic examples include generalized 3-body Hamiltonians at zero energy such as $Δ_x+V_0(x')+V(x)$ where $Δ_x$ is the Laplace operator on $\mathbb{R}^n_x=\mathbb{R}^{n-1}_{x'}\times\mathbb{R}_{x''}$, and $V_0$ and $V$ are potentials with at least inverse quadratic decay: this operator is approximately translati…

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) -- for example the solution operator of a classical elliptic problem -- and $P_{1,T}$ is a realization of an elliptic differential operator $P_1$, having a ray free of eigenvalues. Relative formulas (e.g. for the difference between the constants with two different choices of $P_{1,T}$) have been known for some time and…

纤维尖点算子的Calderón投影

The Calderón Projector for Fibred Cusp Operators

本文将Seeley的Calderón投影定理从紧致流形推广到具有纤维尖点结构的非紧流形，利用$\u03d5$-伪微分演算，适用于局部对称空间及奇异空间上的椭圆算子。

带柱状端的流形上的边值问题和层势

Boundary value problems and layer potentials on manifolds with cylindrical ends

本文通过引入参数依赖的层势方法，克服边界非紧致带来的技术困难，证明了全局非紧致情形下边界积分算子的可逆性，并应用于带边界和柱状端的流形上非齐次Dirichlet问题的适定性及Dirichlet-to-Neumann算子的存在性。

Optimal control of a Vlasov-Poisson plasma by an external magnetic field - The basics for variational calculus

We consider the three dimensional Vlasov-Poisson system that is equipped with an external magnetic field to describe a plasma. The aim of various concrete applications is to control a plasma in a desired fashion. This can be modeled by an optimal control problem. For that reason the basics for calculus of variations will be introduced in this paper. We have to find a suitable class of fields that are admissible for this procedure as they provide unique global solutions of the Vlasov-Poisson system. Then we can define a field-state operator that maps any admissible field onto its corresponding …

An index theorem of Callias type for pseudodifferential operators

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + i Φ, where D is elliptic pseudodifferential with Hermitian symbols, and Φis a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Mel…

Weyl演算视角在有界域离散八元数分析中的应用

Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains

本文利用基于Weyl演算的方法研究$\mathbb{R}^{8}$有界域上的离散八元数分析，证明并推广离散Stokes公式，建立Borel-Pompeiu与Cauchy公式，并阐明非结合性的作用。