Contraction semigroups of elliptic quadratic differential operators

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.     

Deformation quantization and invariant differential operators

In this Note we explain how the techniques of deformation quantization in the sense of Kontsevich can be used to describe the algebra of invariant differential operators on Lie groups.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

On differential operators over a map,thick morphisms of supermanifolds,and symplectic micromorphisms

We recall the notion of a differential operator over a map (in linear and non-linear settings) and consider its versions such as formal ħ-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and operators arising as quantization of symplectic micromorphisms.     

Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps

In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators.     

Existence of natural and projectively equivariant quantizations

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operators, and prove the existence of a quantization except when this parameter belongs to a discrete set of resonant values.     

关于二次Putnam-Fuglede定理(Ⅰ)

设Ｎ为正规算子,若Ｎ与交换子ＮＸ－ＸＮ可交换,则Ｎ必与Ｘ可交换,称此结论为二次Ｐｕｔｎａｍ－Ｆｕｇｌｅｄｅ定理．本文给出了在幂零算子扰动下及在一些非正常算子时的二次ＰＦ定理     

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

Gaussian kernel operators on white noise functional spaces

The Gaussian kernel operators on white noise functional spaces, including second quantization, Fourier-Mehler transform, scaling, renormalization, etc. are studied by means of symbol calculus, and characterized by the intertwining relations with annihilation and creation operators. The infinitesimal generators of the Gaussian kernel operators are second order white noise operators of which the number operator and the Gross Laplacian are particular examples.     

Approximation of solutions to impulsive functional differential equations

In this paper we shall study a semilinear impulsive functional differential equation in a separable Hilbert space. We shall use the analytic semigroups theory of linear operators and fixed point technique to establish the existence, uniqueness, and the convergence of approximate solutions to the given problem. We will also prove the existence and convergence of finite-dimensional approximate solutions to the given problem. An example is also illustrated.     

Zur Konstruktion gewisser Integraloperatoren für partielle Differentialgleichungen Teil II

Using results of Part I of this paper, we shall now develop two methods of constructing linear partial differential equations which admit Bergman operators with polynomial kernels; these equations will be obtained explicitly. Those methods will also yield general representations of solutions of such an equation which are holomorphic in some domain of complex two-space. For generating all those solutions, one needs a pair of Bergman operators. Whereas in Part I of this paper we required at least one of the two operators to have a polynomial kernel, we now impose the condition that both operators be of that kind. This entails further basic results about the existence, construction, and uniqueness of solutions.     

J—自共轭微分算子谱的定性分析

本文对J－自共轭微分算子谱理论研究情况做一些概要性的介绍,第一部分简要回顾了J－自共轭微分算子理论研究的发展过程,第二,三部分介绍了J－自共轭微分算子的本质谱和离散谱定性分析的主要方法和结论;第四部分扼要叙述J－自共轭微分算子其它方面的一些工作,以及J－自共轭微分算子谱理论研究中尚待解决的问题。     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

Quantization of classical mechanics: Shall we lie?

We propose a Lie-Noether-symmetry solution of two problems that arise with classical quantization: the quantization of higher-order (more than second) Euler-Lagrange ordinary differential equations of classical mechanics and the quantization of any second-order Euler-Lagrange ordinary differential equation that classically comes from a simple linear equation via nonlinear canonical transformations.     

关于(s,p)-w-亚正常算子类

本文在W-亚正常算子类的基础上，引入（s，p）-w-亚正常算子类，进而讨论了该类算子的特征，包含关系，对一类特殊的该类算子还考虑了其平方性质．     

Wick Product and Backward Heat Equation

In the present paper the Wick version of analytic functions with respect to a one dimensional Brownian motion is shown to be closely related to the backward heat equation. This fact provides representation theorems for a certain class of random variables in terms of Wick powers. In addition, we obtain explicit formulas for the action of some second quantization operators arising in the applications.     

A universal formula for deformation quantization on Kähler manifolds

We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kähler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.     

广义微分二次量子化算子

该文对任一从 E_c 到 E_c^* 的连续线性算子定义了 其广义微分二次量子化算子, 由Schwartz 核定理得到其Fock 展开,并用张量积的缩合给出复合算子的微分二次量子化算子.     

GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of Koopman operators for chaotic dynamical systems, in terms of annihilation operators dt and its dual, creation operators t*.     

Honest submodules

Lattices of submodules of modules and the operators we can define on these lattices are useful tools in the study of rings and modules and their properties. Here we shall consider some submodule operators defined by sets of left ideals. First we focus our attention on the relationship between properties of a set of ideals and properties of a submodule operator it defines. Our second goal will be to apply these results to the study of the structure of certain classes of rings and modules. In particular some applications to the study and the structure theory of torsion modules are provided.