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We give an explicit formula, as a formal differential operator, for quantummicroformal morphisms of (super)manifolds that we introduced earlier. Suchquantum microformal morphisms are essentially oscillatory integral operators orFourier integral operators of a particular kind. They act on oscillatory wavefunctions, whose algebra extends the algebra of formal power series in Planck'sconstant.  In the classical limit, quantum microformal morphisms reproduce, as the mainterm of the asymptotic, the nonlinear pullbacks of functions with respect to`classical' microformal morphisms. We found these nonlinear transformations offunctions in search of $L_{\infty}$-morphisms for homotopy Poisson structures.

Quantum microformal morphisms of supermanifolds: an explicit formula and further properties