The construction of Gaussian beams satisfying to the wave equation with exponential exactness

A function $V$ having Gaussian beam asymptotic (the frequency $\omega\to+\infty$) is obtained. The expression $(\Delta+\frac{\omega^2}{c^2(x)})V$ is exponentialy small in the the neighbourhood of axial ray of Gaussian beam. The velocity $c(x)$ is analitical.