Properties Of Pseudodifferential Operators Research Articles

Boundary value problems of elliptic operators and reduction to the boundary techniques

We study properties of pseudodifferential operators which arise in their use in boundary value problems. Smooth domains as well as intersections of smooth domains are considered.

Pseudo-differential operators of infinite order on $$W^{\Omega }_{M}(\mathbb C^{n})$$ W M Ω ( C n ) -space involving fractional Fourier transform

The mapping properties of pseudo-differential operators associated with symbol \(\theta (z,\xi ), z= x+iy\) and \(\xi =u+it\) on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space are investigated. Motivated from Wong (An introduction to pseudo-differential operators. World Scientific, Singapore, 2014), \(L^{p}(\mathbb R^{n})\)-boundedness of pseudo-differential operators on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space is studied by using the fractional Fourier transform.

Iterative properties of pseudo-differential operators on edge spaces

Pseudo-differential operators with twisted symbolic estimates play a large role in the calculus on manifolds with edge singularities. We study here aspects of the underlying abstract concept and establish a new result on iteration of quantizations.

Characterization of spaces of type $$W$$ W and pseudo-differential operators of infinite order involving fractional Fourier transform

The characterization of W-type spaces is investigated and various properties of pseudo-differential operators are studied by using the fractional Fourier transform.

The Berezin and Garding Inequalities

Let ϕ be a convex function on ℂ, let ℒ(σ) be a pseudodifferential operator with symbol σ, let Λσ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λσ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that $$\sum {_{\lambda \in \Lambda _\sigma } m(\lambda )} \varphi (\lambda ) \leqslant \operatorname{Tr} L(\varphi (\sigma )) + R$$ , where R is an error term of the same order as the remainder term in the Garding inequality.

On the mathematical theory of the Aharonov$ndash$Bohm effect

We consider the Schr?dinger operator H = (i? + A)2 in the space L2(2) with a magnetic potential A(x) = a()(?x2, x1) |x|?2, where a is an arbitrary function on the unit circle. Our goal is to study spectral properties of the corresponding scattering matrix S(?), ? > 0. We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator of zero order whose symbol is an explicit function of a. We deduce from this result that the essential spectrum of S(?) does not depend on ? and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle. Finally, we calculate the diagonal singularity of the scattering amplitude (kernel of S(?) considered as an integral operator). In particular, we show that for all these properties only the behaviour of a potential at infinity is essential. The preceding papers on this subject treated the case a() = const and used the separation of variables in the Schr?dinger equation in the polar coordinates. This technique does not, of course, work for arbitrary a. From an analytical point of view, our paper relies on some modern tools of scattering theory and well-known properties of pseudodifferential operators.