An inverse problem for an harmonic oscillator perturbed by potential: Uniqueness

摘要

Consider the perturbed harmonic oscillator Ty = -y" + x(2)y + q(x)y on L-2(R), where the real potential q satisfy some assumption on infinity (the case q is an element of L-2 (R, (+ 1)(-r) dt), r < 1 is covered). The spectrum of T is purely discrete and consists of the simple eigenvalues λ(n) &UARR; &INFIN;. Let ψ(n) be the corresponding eigenfunctions. Define the norming constants v(n)(q) = lim(x&UARR;&INFIN;) log ψ(n)(x)/ψ(n)(-x). We prove that the spectrum and the norming constants together uniquely determine the potential. Also, we announce the result about the full characterization of the spectral data in the case q', xq &ISIN; L-2 (R). [References: 12]