Abstract
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.
Highlights
Introduction and Main TheoremsIn this work, we study the inverse scattering problem of recovering a first-order perturbation from fixed angle scattering measurements
We study the inverse scattering problem of recovering a first-order perturbation from fixed angle scattering measurements
We follow the approach used in [19]; that is, we show the equivalence of the fixed angle scattering problem with an appropriate inverse problem for the wave equation
Summary
We study the inverse scattering problem of recovering a first-order perturbation from fixed angle scattering measurements. This problem can be considered in the frequency domain, as the problem of determining q from the scattering amplitude aq( · , · , ω) for the Schrodinger operator −Δ + q with a fixed direction ω ∈ Sn−1, or alternatively in the time domain as the problem of recovering q from boundary or scattering measurements of the solution Uq of the wave equation The equivalence of these problems is discussed in [19] (see [16,17,26] for the odd-dimensional case). In the last section of the paper, we state and prove Theorems 5.1 and 5.2 in order to illustrate how the number of measurements can be reduced in time domain by imposing symmetry assumptions on the potentials. (Theorem 1.3 follows from the second result.) The proof of Theorem 1.4 given in “Appendices A and B” is devoted to adapting several known results for the wave operator to our purposes
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