Tunneling and Agmon estimates

For P_h=−h²Δ/2+V with real smooth V, take a compactly supported smooth test v and a real smooth weight φ. Integration by parts gives the identity below, with no boundary term because of compact support. The gradient of the weight subtracts |∇φ|²/2 from the effective potential. This factor one half follows from the mass-one kinetic normalization; using a distance formula from a source with kinetic term −h²Δ without rescaling would give the wrong decay exponent.

For a normalized eigenfunction, one applies the identity to a cutoff times that eigenfunction. The cutoff creates controlled source terms, so those terms must be retained rather than declaring that every weighted energy term vanishes. In a region where V−E−|∇φ|²/2 is positive, the identity controls the weighted amplitude and gradient there using contributions from the allowed region and the cutoff. This is an energy proof of decay, distinct from assuming a pointwise WKB ansatz is globally valid.

For a fixed energy E, the Agmon length of a path integrates sqrt(2(V−E)_+) against ordinary arclength. Taking the infimum from the allowed set V≤E defines an Agmon distance. It satisfies the corresponding eikonal inequality almost everywhere, and a slightly reduced weight (1−ε) times that distance leaves a positive margin in the forbidden region. Smooth approximations and localization justify using this generally Lipschitz weight in the energy argument. Constants must be controlled over any energy interval being considered.

Under the standard confining self-adjoint Schrödinger hypotheses and an appropriate separated forbidden region, these weighted estimates yield exponential decay up to specified constants or polynomial h losses. The distance uses the relevant eigenenergy, which may depend on h. A bottom-energy action can instead describe a leading exponential scale for low-lying states, but it must not be mistaken for an exact zero-energy eigenfunction. The kinetic energy usually raises the true ground energy above the minimum of V.

In a symmetric smooth double well with nondegenerate minima, low-lying states can be built near the individual wells and coupled through the barrier. Agmon estimates give small overlaps and useful upper bounds. A sharp leading splitting additionally needs a controlled interaction matrix, nonvanishing coupling and suitable barrier geometry; in higher dimensions multiple paths can complicate the coefficient. Symmetry, well equivalence and the chosen spectral pair must be stated before interpreting a tiny difference as tunneling.

It is therefore unsafe to write splitting equals exp(−distance/h) as a universal identity. The exponent may encode the leading action, while h powers and multiplicative constants depend on the well and barrier structure. A decay estimate alone supplies no matching lower bound and can miss cancellations. The elementary barrier calculation below deliberately concerns a local decaying solution, so its exact exponential attenuation is not advertised as a full transmission coefficient or a double-well spectral theorem.