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 $b\lambda$, the lower bounds have the form $e^{-b\lambda d_1^2-C{d}_1^{1+\delta }}$ for every fixed $\delta \in (0,1)$. 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.