Analytic description of inversion vibrational mode for ammonia molecule

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 $NH_3$.