Inverse Scattering with Magnetic Potentials

• Inverse scattering with magnetic potentials is a study of reconstructing electromagnetic fields from scattering data, emphasizing gauge equivalence and asymptotic analysis.
• Methodologies involve analyzing Schrödinger operators with long‐range magnetic potentials, using potential decomposition and gauge transformation techniques.
• Applications span quantum imaging and the Aharonov–Bohm effect, providing insights into the reconstruction and uniqueness of electromagnetic field configurations.

Inverse scattering with magnetic potentials refers to the problem of determining information about electromagnetic fields—most notably the magnetic vector potential—from measurable scattering data produced by quantum particles or electromagnetic waves interacting with material systems. This topic encompasses rigorous analysis of the Schrödinger operator (with magnetic and electric potentials) in various geometric settings, the associated gauge structure, the connection between scattering observables and the underlying potentials, and the mathematical methods for reconstructing those potentials (up to gauge equivalence) from asymptotic data.

1. Gauge Equivalence and Its Role in Scattering Theory

A defining feature of magnetic Schrödinger operators is the gauge freedom in the magnetic potential $A(x)$. Two potentials $A(x)$ and $A'(x)$ are gauge equivalent if there exists a function $g(x)$ in the gauge group $G(\Omega)$ such that $A'(x) = A(x) - i g^{-1}(x) \nabla g(x)$, and the Schrödinger operator $(–i\nabla – A(x))^2 + V(x)$ transforms accordingly through $u(x) \mapsto g(x)u(x)$ (see Eq. (1.11)). The gauge group $G(\Omega)$ is characterized by functions that asymptotically behave like $\exp[i(m\theta + L(x))]$, with $L(x)$ real-valued and decaying appropriately, and $m \in \mathbb{Z}$ (see Eq. (1.9) and decay condition (1.7)). Gauge transformations induce explicit phase shifts in the scattering matrix—the $S$-matrix—via conjugation with exponential operators involving derivatives (see Eqs. (1.12), (1.13)).

In inverse scattering, physical observables depend only on the gauge class of $A(x)$, so reconstruction from scattering data must always be modulo gauge transformations. The challenge is thus to identify the gauge equivalence class of the operator $(A(x), V(x))$ from the gauge equivalence class of the $S$-matrix.

2. Structure and Decay of Magnetic and Electric Potentials

The model under analysis involves the Schrödinger operator

$H = (–i\nabla – A(x))^2 + V(x)$

in a domain $Q = \mathbb{R}^n \setminus O$ where $O$ is a smooth bounded obstacle ($n \geq 2$). The magnetic potential $A(x)$ is assumed long-range, with higher-order derivatives decaying at infinity as $$|\nabla^\alpha A(x)| \leq C \langle x \rangle^{–1–\varepsilon}$$ for some $\varepsilon > 0$ (see Eqs. (1.2), (1.3)). The electric potential $V(x)$ is short-range, decaying faster than any inverse power of $|x|$.

Outside a large ball, $A(x)$ admits a canonical decomposition (Eq. (1.5)) as $A(x) = A_0(x) + A_1(x)$, with $A_0(x)$ homogeneous of degree $–1$ and satisfying the transversality condition $x \cdot A_0(x) = 0$ (Eq. (1.6)). In dimension $n = 2$, $A_0(x)$ is gauge equivalent to an Aharonov–Bohm-type potential, exhibiting the essential physics of magnetic fluxes concentrated in thin tubes.

3. Correspondence Between Gauge Classes of Hamiltonians and Scattering Matrices

A central result is a one-to-one correspondence between gauge equivalence classes of Hamiltonians $(A(x), V(x))$ and those of $S$-matrices $S(\lambda)$—the scattering matrices at energy $\lambda$. Explicitly, for $n \geq 3$ the transformation law

$$S'(\lambda) = e^{i\psi(D_x)} S(\lambda) e^{-i\psi(-D_x)}$$

holds (Eq. (1.12)), while in $n = 2$ additional phase factors from the gauge group enter (Eq. (1.13)). This correspondence ensures that if two operators yield gauge equivalent $S$-matrices, their potentials must be gauge equivalent as well.

This result has powerful implications: the scattering matrix “remembers” the complete information about the electromagnetic fields modulo gauge. Conversely, under geometric and decay hypotheses (exterior to a convex obstacle, prescribed decay), matching scattering matrices entail matching gauge classes of the underlying potentials.

4. Geometric Aspects: Exterior Domains and Convex Obstacles

The analysis is conducted in domains exterior to bounded convex obstacles, $\Omega = \mathbb{R}^n \setminus O$. The convexity and smoothness of $O$ are critical for two reasons:

• They guarantee simple connectivity (especially when $n \geq 3$), which simplifies gauge considerations and ensures uniqueness up to gauge.
• They allow the use of straight lines (avoiding the obstacle) in integrals relevant to the uniqueness proofs, especially in reconstructing the potential from its scattering data.

Convex obstacles facilitate application of geometric techniques to show that scattering data can be linked directly to line integrals of potential differences, ultimately leading to unique reconstruction modulo gauge.

5. Analytical Formulations and Inverse Reconstruction Process

Key equations and strategies include:

• The gauge transformation law $A'(x) = A(x) - i g^{-1}(x) \nabla g(x)$.
• Asymptotic decomposition: $A(x) = A_0(x) + A_1(x)$, $x \cdot A_0(x) = 0$.
• Scattering matrix transformations for $n \geq 3$ and $n = 2$.
• For two-dimensional cases, potentials of the form

$$A(x) = \frac{(-x_2, x_1)}{|x|^2}(a + \ldots) + A_1(x)$$

are gauge equivalent to Aharonov–Bohm configurations, with $a(\theta)$ such that $\int_0^{2\pi} a(\theta) d\theta = 0$.

In practice, the inverse problem proceeds by analyzing the action of the $S$-matrix on the asymptotic boundary data, applying the explicit gauge transformation formulas, and leveraging geometric conditions to ensure recoverability up to gauge.

6. Physical Implications and Applications

The established correspondence brings several consequences:

• Full information about both the magnetic and electric fields can be recovered from the $S$-matrix, up to gauge.
• The approach is directly relevant for the quantum Aharonov–Bohm effect, demonstrating how magnetic fluxes affect scattering phenomena even when the magnetic field vanishes locally.
• The methods support applications in quantum and electromagnetic imaging, where macroscopic measurable data are used to infer internal field information.
• Extensions of the approach may reach settings such as hyperbolic inverse problems, mathematical physics, and engineering domains requiring unique identification of electromagnetic configurations.

A plausible implication is that similar correspondence principles may hold for broader classes of operators or obstacles, provided analogous geometric and decay conditions apply.

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The paper of inverse scattering with magnetic potentials thus rigorously establishes how gauge structures and asymptotic properties of potentials dictate the information encoded in scattering data, delineates the constraints and strategies for their recovery, and elucidates the interplay of geometry, analysis, and physics in the formation and interpretation of observable scattering phenomena (Eskin et al., 2011).