We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.

我们研究了从对应于有限多个平面波的散射测量确定磁场和电势的逆散射问题。主要结果表明，系数由 2 n 次测量唯一确定，直到自然规范。我们还表明，可以恢复没有规范不变性的相关方程的完整一阶项，并且如果系数具有某些对称性，则可以减少测量次数。这项工作将 Rakesh 和 Salo (SIAM J Math Anal 52(6):5467–5499, 2020) 和 (Inverse Probl 36(3):035005, 2020) 的固定角散射结果扩展到具有一阶扰动的哈密顿量，以及它基于波动方程方法和卡尔曼估计。