Finite-energy electron Airy beams in a uniform magnetic field are investigated by reducing the transverse dynamics to a shifted one-dimensional harmonic oscillator. For an exponentially truncated Airy beam, an exact Airy-function representation of the propagated wave function is derived, and its asymptotic behavior at the characteristic moments of the magnetic period is examined. Near odd quarter periods, the Airy argument is driven toward the oscillatory sector, and direct evaluation of the Airy representation is therefore numerically delicate. To avoid this numerical difficulty, we further derive an exact Wigner representation, in which the dynamics appears as a rigid harmonic rotation in phase space and the position and momentum marginals remain stable throughout the magnetic period. For obstructed beams, the phase-space formulation is combined with a phase-sensitive similarity measure in a finite window. The analysis separates the Gaussian-amplitude transition, in which the state acquires a Gaussian envelope differing from the initial Airy state, from exact oscillator recurrence and uses Wigner regularization as a stable phase-space diagnostic; as an application, local self-healing is demonstrated for obstructed beams. The results show that the obstruction is not globally erased by the unitary evolution, but the surviving Airy components can locally reconstruct the beam profile within selected observation windows. This work provides a phase-space framework for evaluating structured electron beams in magnetic fields and offers a quantitative diagnostic for local self-healing in magnetically guided Airy beams.

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• Particle phenomena
• Path integrals
• Particle beams