Non-commutative normal form, spectrum and inverse problem

Article type: Research Article

Authors: Anikin, Anatoly

Affiliations: A. Ishlinski Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow Institute of Physics and Technology, Bauman Moscow State Technical University, Prosp. Vernadskogo 101-1, 119526 Moscow, Russia. E-mail: anikin83@inbox.ru

Abstract: We propose an approach to calculate asymptotic series for low lying eigenvalues of Schrödinger operator based on normal forms in the formal graded Weyl–Heisenberg algebra. The difference from a traditional scheme is that we don’t use any symbol map (Weyl, px, xp, etc.). We show that our method may be useful for different reasons. Firstly, it enables to estimate the growth of the eigenvalues expansion coefficients, and secondly it may be efficient for practical calculations, e.g. for treating inverse problems. In particular, we prove that under some restrictions in the one-dimensional case the knowledge of asymptotic series for any pair of low lying eigenvalues is enough to recover the potential.

Keywords: Schrödinger operator, spectrum, Birkhoff normal form, inverse problem

DOI: 10.3233/ASY-161399

Journal: Asymptotic Analysis, vol. 101, no. 4, pp. 207-225, 2017