  Motivated by the analysis of the tunneling effect for the magnetic Laplacian,
we introduce an abstract framework for the spectral reduction of a self-adjoint
operator to a hermitian matrix. We illustrate this framework by three
applications, firstly the electro-magnetic Laplacian with constant magnetic
field and three equidistant potential wells, secondly a pure constant magnetic
field and Neumann boundary condition in a smoothed triangle, and thirdly a
magnetic step where the discontinuity line is a smoothed triangle. Flux effects
are visible in the three aforementioned settings through the occurrence of
eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with
double well radial potential, we rule out an artificial condition on the
distance of the wells and extend the range of validity for a recently
established tunneling approximation, thereby settling the problem of
electro-magnetic tunneling under constant magnetic field and a sum of
translated radial electric potentials.
