A proof of exponential suppression of high-energy transitions in the anharmonic oscillator

Abstract

We derive rigorous bounds on the matrix elements of the position operator between a low-lying and an arbitrary excited state of the quartic anharmonic oscillator. For both the single-and the double-well potentials, these bounds decrease monotonically and exponentially fast with the energy difference between these states, measured in units of the perturbative frequency. The order of the exponential decay changes smoothly from 1 to 3 4 at some non-perturbative large energy scale. Our bounds prove that transitions, induced by an external infinitesimal but rapidly-oscillating force, never become strong contrary to the predictions of the tree-level and leading-order instanton calculations. We explain why these induced transitions are the quantum-mechanical analog of two-particle collision processes with large multiplicity in the final state.