Asymptotic inverse spectral problem for anharmonic oscillators

Abstract

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(−∂²+x ²−1) on ℝ, when potentialB(x) has a prescribed asymptotics at ∞,B(x)∼|x|^−α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^−γ), γ=1/2+1/4.

The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k },

$$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$

whose leading term$\tilde V$ represents the so-called “Radon transform” ofV,

$$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$

as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^−α|V(x) ofB from asymptotics of {µ _k }. ^∞₁