Asymptotic inverse spectral problem for anharmonic oscillators with odd potentials

Abstract

The present paper continues our earlier work on asymptotic spectral problems for perturbations of the harmonic oscillator L =A + B on R, where A = ½(-∂^2 + x^2). The class of functions B is defined by their asymptotics at {∞} B(x)~ │x│^(-ɑ) x trigonometric V(x). The even-potential case of the author's earlier paper, i.e. V=Σ ɑ_m cos ω_mx, is now extended to a more difficult odd case, V = Σ b_m sin ω_mx. Precisely, the eigenvalues of L, λ_k = k + µ_k are shown to admit an asymptotic expansion µ_k ~ constant xk^(-y)V(√2k+... as k → ∞ where the exponent y = ½ ɑ + ¾, and V denotes the 'odd version' of the so-called 'Radon transform' of V. As a consequence we get a unique and explicit solution of the asymptotic inverse spectral problem for odd perturbations of A, similar to the even case previously consider by the author. The methods employed here combine some ideas of the n-sphere Schrödinger theory as well as the techniques developed by the author.