相关论文: Schwartz regularity of differential operators on t…
A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is…
We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x),…
We introduce a class of time-periodic Gelfand-Shilov spaces of functions on $\mathbb{T} \times \mathbb{R}^n$, where $\mathbb{T} \sim \mathbb{R} /2\pi \mathbb{Z}$ is the one-dimensional torus. We develop a Fourier analysis inspired by the…
In this paper, we investigate the global properties of Fourier multipliers in the setting of nonharmonic analysis of boundary value problems. We give necessary and sufficient conditions for a Fourier multiplier to be globally hypoelliptic…
This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system…
It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential…
The aim of this paper is to give a review of local and global properties of Fourier integral operators with real and complex phases, in local $L^p$, global $L^2$, and in Colombeau's spaces.
Differential operators on Schwartz distributions conventionally are defined as the transpose of differential operators on functions with compact support. They do not exhaust all differential operators. We follow algebraic formalism of…
This article presents an investigation on the global hypoellipticity problem for systems belonging to the class $P = D_t + Q(t,D_x)$, where $Q(t,D_x)$ is a $m\times m$ matrix with entries $c_{j,k}(t)Q_{j,k}(D_x)$. The coefficients…
The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in \mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which…
We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the…
We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose…
We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly…
We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans…
The local $L^2$-mapping property of Fourier integral operators has been established in H\"ormander \cite{H} and in Eskin \cite{E}. In this paper, we treat the global $L^2$-boundedness for a class of operators that appears naturally in many…
Euler operators are partial differential operators of the form $P(\theta)$ where $P$ is a polynomial and $\theta_j = x_j \partial/\partial x_j$. They are surjective on the space of temperate distributions on $R^d$. We show that this is, in…
The composition of the Fourier transform in $\mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on…
In this paper we develop the generalised Schur theory offered in the recent paper by the second author in dimension one case, and apply it to obtain a new explicit parametrisation of torsion free rank one sheaves on projective irreducible…