I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \mathbb{R}$, $$ F(x - a) e^{x a} + (1 - F(x+a)) e^{-x a} = e^{a^2/2}. $$ I had never seen anything like that before. Any references are most welcome.