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Abstract

We study the two--dimensional magnetic Schr\"odinger operator with a penetrable circular wall modeled by a $\delta$--interaction. Using the boundary triple approach we classify all self--adjoint extensions and obtain Krein's resolvent formula, showing that the essential spectrum coincides with the Landau levels. The wall breaks their infinite degeneracy and produces a spectral broadening: each Landau level becomes an accumulation point of discrete eigenvalues from one side. In the circular case, rotational symmetry reduces the eigenvalue problem to scalar equations with explicit Weyl coefficients. We prove strict monotonicity, ensuring that each angular momentum channel contributes at most one eigenvalue per gap, and derive asymptotics showing that the boundary coefficients decay faster than any exponential, explaining the strong localization of the broadened spectrum. Numerical simulations are consistent with these results.