Tunneling contributions to matrix elements of the anharmonic oscillator

We study the double-well anharmonic oscillator when the barrier between the two wells is broad and high. In this limit, the energy gap between even- and odd-parity eigenstate pairs is exponentially small. We develop equations for similar exponentially small tunneling contributions to arbitrary matrix elements of arbitrary operators. As an example, we calculate the matrix elements $〈m\pm \left|x^{2l}\right|n\pm 〉$ and $〈m\pm \left|x^{2l+1}\right|n\mp 〉$.