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Based on Johnson's operator formula for the equivariant Gromov-Witten theoryof $\mathbb{P}^1$-orbifolds, we give a new approach to the operator formalismby Okounkov and Pandharipande regarding the $\mathbb{C}^*$-equivariantGromov-Witten theory of $\mathbb{P}^1$ relative to one or two points. We alsoextend their operator formalism in the non-equivariant specialization to thecase where we allow negative contact orders.

A new approach to the operator formalism for Gromov-Witten invariants of the cap and tube