Communications in Mathematical Physics

, Volume 249, Issue 1, pp 133–196

Inverse Problem for Harmonic Oscillator Perturbed by Potential, Characterization

Article

DOI: 10.1007/s00220-004-1105-8

Cite this article as:
Chelkak, D., Kargaev, P. & Korotyaev, E. Commun. Math. Phys. (2004) 249: 133. doi:10.1007/s00220-004-1105-8

Abstract

Consider the perturbed harmonic oscillator Ty=-y’’+x2y+q(x)y in L2(ℝ), where the real potential q belongs to the Hilbert space H={q’, xqL2(ℝ)}. The spectrum of T is an increasing sequence of simple eigenvalues λn(q)=1+2nn, n ≥ 0, such that μn→ 0 as n→∞. Let ψn(x,q) be the corresponding eigenfunctions. Define the norming constants νn(q)=limx↑∞log |ψn (x,q)/ψn (-x,q)|. We show that Open image in new window for some real Hilbert space Open image in new window and some subspace Open image in new window Furthermore, the mapping ψ:q↦ψ(q)=({λn(q)}0, {νn(q)}0) is a real analytic isomorphism between H and Open image in new window is the set of all strictly increasing sequences s={sn}0 such that Open image in new window The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, pL2(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces Open image in new window We obtain their basic properties, using their representation as spaces of analytic functions in the disk.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Dmitri Chelkak
    • 1
  • Pavel Kargaev
    • 2
  • Evgeni Korotyaev
    • 3
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Faculty of Math. and MechSt-Petersburg State UniversitySt. PetersburgRussia
  3. 3.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany

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