切片超复正则性与 $S$-解析算子及边界条件

The $\mathcal{F}$-resolvent equation and Riesz projectors for the $\mathcal{F}$-functional calculus

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collabo…

Non commutative functional calculus: bounded operators

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.

A direct approach to the S-functional calculus for closed operators

The $S$-functional calculus for slice hyperholomorphic functions generalizes the Riesz-Dunford-functional calculus for holomorphic functions to quaternionic linear operators and to $n$-tuples of noncommuting operators. For an unbounded closed operator, it is defined, as in the classical case, using an appropriate transformation and the $S$-functional calculus for bounded operators. This is however only possible if the $S$-resolvent set of the operator contains a real point. In this paper, we define the $S$-functional calculus directly via a Cauchy integral, which allows us to consider also ope…

Properties of a polyanalytic functional calculus on the $S$-spectrum

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by $\mathcal{D}$. This theorem is divided in two steps. In the first step a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for these type of functions is the starting point of the $S$-functional calculus. In the second step a monogenic function is obtained by applying the Laplace operator in four real variables, namely $ Δ$, to a sl…

非自治哈密顿演化方程的谱截断振荡积分

Spectral Cut-off Oscillatory Integrals for Non-Autonomous Hamiltonian Evolution Equations

针对非自治哈密顿演化方程，通过谱截断构造有限维振荡积分，证明其收敛到原方程强解，并联系Floquet-Magnus展开。