[Paper Review] The inverse problem for perturbed harmonic oscillator on the half-line

Dmitry Chelkak, Evgeny Korotyaev|ArXiv.org|Aug 29, 2005

Spectral Theory in Mathematical Physics13 references3 citations

TL;DR

This paper solves the inverse spectral problem for the perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions, proving that the spectral data—comprising eigenvalues and a modified set of norming constants—uniquely determine the potential $ q \in \mathbf{H}_+ $. The key contribution is the introduction of transformed coordinates $ \{r_{2n+1}\} \in \ell^{2}_{3/4} $ that serve as independent, real-analytic coordinates for the potential space, enabling a complete characterization of admissible spectral data.

ABSTRACT

We consider the perturbed harmonic oscillator $T_Dψ=-ψ''+x^2ψ+q(x)ψ$, $ψ(0)=0$ in $L^2(R_+)$, where $q\in H_+=\{q', xq\in L^2(R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{ m spectral data}={ m \{eigenvalues of\}T_D{ m \}}\oplus{ m \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in H_+$ is given. Moreover, we solve the similar inverse problem for the family of boundary conditions $ψ'(0)=b ψ(0)$, $b\in R$.

Motivation & Objective

• To solve the inverse problem for the perturbed harmonic oscillator $ T_D = -\psi'' + x^2\psi + q(x)\psi $ on $ \mathbb{R}_+ $ with Dirichlet boundary condition $ \psi(0) = 0 $.
• To characterize the set of spectral data—eigenvalues and norming constants—that correspond to potentials in the class $ \mathbf{H}_+ = \{ q : q', xq \in L^2(\mathbb{R}_+) \} $.
• To establish a one-to-one and onto mapping between $ \mathbf{H}_+ $ and the spectral data, including a novel transformation of norming constants to ensure independence and integrability.
• To extend the inverse spectral theory to the family of operators with Robin-type boundary conditions $ \psi'(0) = b\psi(0) $, $ b \in \mathbb{R} $.

Proposed method

• Introduce a modified set of norming constants $ \{r_{2n+1}\} $ via a transformation of the original $ \{s_{2n+1}\} $, ensuring they lie in $ \ell^{2}_{3/4} $.
• Use asymptotic analysis of eigenfunctions and eigenvalues of the unperturbed harmonic oscillator to derive perturbative expansions in terms of Fourier coefficients of $ q $.
• Apply a spectral mapping approach that treats $ \{\mu_{2n+1}(q)\} $, $ q(0) $, and $ \{r_{2n+1}(q)\} $ as independent coordinates in the potential space.
• Establish real-analytic isomorphism between $ \mathbf{H}_+ $ and the product space $ \mathcal{S}_D \times \mathbb{R} \times \ell^{2}_{3/4} $, ensuring bijectivity and smoothness.
• Prove convergence of critical double sums in the perturbation expansion using $ L^2 $-norm estimates and asymptotic decay of eigenfunction products.
• Generalize the inverse problem to the family $ \{T_b\}_{b \in \mathbb{R}} $, deriving an explicit formula for $ b $ in terms of spectral data (Theorem 2.5).

Experimental results

Research questions

• RQ1 Can the potential $ q \in \mathbf{H}_+ $ be uniquely reconstructed from its Dirichlet eigenvalues and norming constants?
• RQ2 What is the complete characterization of spectral data corresponding to $ q \in \mathbf{H}_+ $, including eigenvalues and additional data?
• RQ3 How can the norming constants be redefined so that they become independent coordinates in the space of potentials?
• RQ4 Is the inverse spectral mapping from $ \mathbf{H}_+ $ to spectral data a real-analytic isomorphism?
• RQ5 Can the inverse problem be extended to the full family of Robin-type boundary conditions $ \psi'(0) = b\psi(0) $, and if so, how is $ b $ determined from spectral data?

Key findings

• The mapping $ q \mapsto \{\text{eigenvalues of } T_D\} \cup \{\text{norming constants}\} $ is one-to-one and onto for $ q \in \mathbf{H}_+ $.
• The standard norming constants $ \{s_{2n+1}\} $ do not form a suitable coordinate system due to lack of independence and integrability; a transformed set $ \{r_{2n+1}\} $ is introduced.
• The transformed coordinates $ \{r_{2n+1}\} $ lie in $ \ell^{2}_{3/4} $, ensuring they are well-behaved and independent of the eigenvalues.
• The spectral mapping $ q \mapsto \left(\{\mu_{2n+1}(q)\}, q(0), \{r_{2n+1}(q)\}\right) $ is a real-analytic isomorphism between $ \mathbf{H}_+ $ and $ \mathcal{S}_D \times \mathbb{R} \times \ell^{2}_{3/4} $.
• The inverse problem for the family $ \{T_b\}_{b \in \mathbb{R}} $ is solved, with an explicit formula for $ b $ in terms of spectral data (Theorem 2.5).
• The convergence of a critical double sum $ S_k \to 0 $ as $ k \to \infty $ is rigorously proven, which is essential for the perturbation analysis and spectral mapping.