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Sfb288 logo Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 573


Inverse problem for harmonic oscillator perturbed by potential, characterization

Dmitri Chelkak, Pavel Kargaev, and Evgeni Korotyaev

Consider the perturbed harmonic oscillator $Ty=-y''+x^2y+q(x)y$ in $L^2(R)$, where the real potential $q$ belongs to the Hilbert space $H={q', xqin L^2(R )}$. The spectrum of $T$ is an increasing sequence of simple eigenvalues $l_n(q)=1+2n+m_n$,, $nge 0$ such that $m_n o 0$ as $n o iy$. Let $p_n(x,q)$ be the corresponding eigenfunctions. Define the norming constants $ _n(q)=lim{}_{xuaiy}log |p_n (x,q)/p_n (-x,q)|$,. We show that ${m_n}_0^iyin olinebreak cH$,, ${ _n}_0^iyin cH_0$ fo r some real Hilbert space $cH$ and subspace $cH_0ss cH$,. % described as the spaces of analytic functions in the unit disk. Furthermore, the mapping $P:qmapstoP(q)=({l_n(q)}_0^iy, { _n(q)}_0^iy )$ is a real analytic isomorphism between $H$ and $cS scH_0$,, where $cS$ is the set of all strictly increasing sequences $s={s_n}_0^iy$ of the form $s_n=1+2n+c_n$,, ${c_n}_0^iyincH$,. 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 $-y''+py$, $pin L^2(0,1)$ with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces $cH$ and $cH_0$,. We obtain their basic properties, using their representation as spaces of analytic functions in the disk.


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