A phase-integral perspective on $\alpha$-decay

This paper applies the phase-integral method to the stationary theory of $\alpha$-decay. The rigorous form of the connection formulas, and their one-directional nature that was not widely known in the physical literature, are applied. The condition for obtaining s-wave metastable states affects the stationary state at large distance from the nucleus, which is dominated by the cosine of the phase integral minus $\frac{\pi }{4}$. Accurate predictions for the lowest s-wave metastable state and mean life of the radioactive nucleus are obtained in the case of uranium. The final part of the paper describes the phase-integral algorithm for evaluating stationary states by means of a suitable choice of freely specifiable base function. Within this framework, original approximate formulas for the phase integrand with arbitrary values of the angular momentum quantum number are obtained.