Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrödinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrödinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.