  The one-dimensional Schr\"odinger equation with symmetric trigonometric
double-well potential is exactly solved via angular prolate spheroidal
function. Although it is inferior compared with multidimensional counterparts
and its limitations are obvious nevertheless its solution is shown to be
analytic rather than commonly used numerical or approximate semiclassical (WKB)
one. This comprises the novelty and the merit of the present work. Our exact
analytic description of the ground state splitting can well be a referee point
for comparison of the accuracy of numerous WKB formulas suggested in the
literature. The approach reasonably well suits for the inversion mode in the
ammonia molecule $NH_3$ and thus yields a new theoretical tool for its
description. The results obtained provide good quantitative description of
relevant experimental data on microwave and IR spectroscopy of $NH_3$.

Chemical Physics;Mesoscale and Nanoscale Physics