Asymptotics of the eigenvalues of the rotating harmonic oscillator
The asymptotic theory of the radial Schrödinger equation \[ {d^2 w\over dr^2}= \left({(1-r)^2 \over 4\alpha^2}- {\lambda+ {1\over 2} \over \alpha}+ {l(l+1) \over r^2} \right)w \] was developed to find an approximation for the eigenvalues of \(\lambda_N\), which are recessive at both \(r=0\) and \(r=+ \infty\). If there the coupling parameter \(\alpha\) is small, asymptotic expansions are given for \(\lambda\). This approximation consists of two components: an asymptotic expansion in powers of \(\alpha\) and a single term which is exponentially small. The Liouville transformation is required to derive the approximation, the Liouville-Green approximation for an expansion in unbounded domains, the effect of the exponentially small term, and an approximation valid at \(r=0\) (the modified Bessel function is used) are described.
scientific article; zbMATH DE number 1326349