Average lower bound on magnetic double-well tunneling with asymptotically large well separation

We study tunneling between two identical, smooth, compactly supported potential wells in a constant perpendicular magnetic field, with the coupling and magnetic field fixed and the well separation tending to infinity. For a sufficiently small positive magnetic field, we prove lower bounds on the hopping coefficient and on the splitting of the two lowest eigenvalues outside a set of separations of density zero. If the well centers are separated by 2d 1 and the magnetic field strength is bλ, the lower bounds have the form e-bλd 1 2-Cd 1 1+δ for every fixed δ in(0,1). No radial or reflection symmetry of the single-well potential is assumed. The proof combines the explicit Landau resolvent kernel with a complex analytic estimate for logarithmic averages of the hopping coefficient.