Semi-classical-Fourier-integral-operator-valued pseudodifferential operators and scattering in a strong magnetic field
type: Item
instance of: scholarly article
title: Semi-classical-Fourier-integral-operator-valued pseudodifferential operators and scattering in a strong magnetic field
zbMATH Open document ID: 1406.35211
author: Ivana Alexandrova
published in: The Journal of Geometric Analysis
publication date: 2019-01-16T00:00:00Z
review text: Let \(n\geq 3\), \(b>0\), and \((x,y,z)=(x_1,...,x_d,y_1,...,y_d, z)\in {\mathbb R}^d \times {\mathbb R}^d \times {\mathbb R}^{n-2d}\). The author considers the following Schrödinger operator with a strong magnetic and a strong electric fields \(H(b)=h_0(b)+bV(x,y,z)\), where \[ H_0(b)=\sum\limits_{j=1}^d \left[\left(D_{x_j} -\frac{b\mu_j}{2}y_j\right)^2 + \left(D_{y_j} -\frac{b\mu_j}{2}x_j\right)^2\right] -\Delta_z, \mu_j>0. \] It is known that the wave operators for the pair \((H_0(b), H(b))\) exist and are complete. Let \(\mathcal{T}(E,b)\) be the scattering amplitude at energy \(E\). The microlocal structure of the semi-classical scattering amplitude \(\mathcal{T}(\lambda b,b)\) is investigated for non-trapping \(\lambda > 0\). The author develop a framework and establish some of the properties of semi-classical-Fourier-integral-operator-valued pseudodifferential operators and prove that \(\mathcal{T}(\lambda b,b)\) is given by such an operator. Discussion of the two special cases of having the potential \(V\) depend only on the variable \(z\) and having the total dimension equal to \(n = 2d + 1\) is included.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 7002650
zbMATH Keywords: scattering theory
zbMATH Keywords: strong magnetic field
zbMATH Keywords: semi-classical Fourier integral operators
MaRDI profile type: Publication
full work available at URL: s12220-017-9932-y
OpenAlex ID: W2762201643
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DOI: 10.1007/S12220-017-9932-Y
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description: scientific article; zbMATH DE number 7002650