Asymptotic inverse spectral problem for anharmonic oscillators (Q1099009)

The paper studies the direct and inverse spectral problem for perturbations \(L=A+B\) of the harmonic oscillator \(A=()(-\partial^ 2+x^ 2)\) on \({\mathbb{R}}\), where potential B(x) has a prescribed asymptotics at \(\{\infty \}\), \(B(x)\sim | x|^{-\alpha}V(x)\), with a trigonometric function \(V(x)=\sum a_ m \cos \omega_ mx.\) The k-th eigenvalue of L is shown to be \(\lambda_ k=k+\mu_ k\) with \(\mu_ k=O(k^{-\gamma})\), \(\gamma =(\alpha /2)+(1/4)\). Furthermore, precise asymptotics of spectral shifts \(\{\mu_ k\}\) are found in terms of the potential function B, \(\mu_ k\sim k^{-\gamma}\tilde V(\sqrt{2k})\), where \(\tilde V\) is the so called ``Radon transform'' of V, \(\tilde V=Const\sum (a_ m/\sqrt{\omega_ m})\cos (\omega_ mx-(\pi /4)).\) These formulae lead to a unique and explicit solution of the inverse problem, i.e. reconstruction of the asymptotic potential data (\(\alpha\),V) from asymptotics of spectral shifts.