Singular star-exponential functions

Abstract

The $\ast$-exponential function $e_{\ast }^{ti{e}^{av}\ast u}$ is defined in a transcendently extended Weyl algebra. In the Weyl ordering expression, this is given as the real analytic solution in $t$ of

$${\partial }_t{F}_t(u,v)=i{e}^{av}\{(u+\frac{\hslash ia}{2})F_t(u+\frac{\hslash ia}{2},v)-\frac{\hslash i}{2}{\partial }_v{F}_t(u+\frac{\hslash ia}{2},v)\},F_0(u,v)=1.$$.

For generic initial functions, this equation can be solved for all $t$, but the uniqueness holds only for one direction.