A smooth counterexample to critical--subcritical heat-kernel comparison

We construct two nonnegative Schrödinger operators on the line with smooth bounded potentials and spectral bottom zero. One is subcritical and the other null-critical, while the quotient of their diagonal heat kernels has infinite limsup. This gives a negative answer to the heat-kernel comparison question without an ordering assumption on the potentials. The construction preserves small heat-kernel values of successive operators with positive spectral gaps, then restores criticality by adding distant intervals on which a positive zero-energy solution equals one.