Semi-classical-Fourier-integral-operator-valued pseudodifferential operators and scattering in a strong magnetic field
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.
semi-classical Fourier integral operators
scientific article; zbMATH DE number 7002650
scientific article