[Paper Review] Reconstruction of energy-dependent Sturm-Liouville equations from two spectra

Nataliya Pronska|arXiv (Cornell University)|May 21, 2012

Spectral Theory in Mathematical Physics23 references4 citations

TL;DR

This paper presents a constructive algorithm for reconstructing energy-dependent Sturm-Liouville operators from two spectra—Dirichlet and mixed boundary conditions—by reducing the problem to inverse spectral analysis of Dirac operators with special potentials. The key contribution is proving existence and uniqueness of the reconstruction under minimal smoothness assumptions, including distributions and singular potentials.

ABSTRACT

In this paper we study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from two spectra. We give a reconstruction algorithm and establish existence and uniqueness of reconstruction. Our approach essentially exploits the connection between the spectral problems under study and those for Dirac operators of a special form.

Motivation & Objective

• To solve the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from two given spectra under minimal regularity assumptions on the potentials.
• To establish the existence and uniqueness of the reconstruction for energy-dependent Sturm-Liouville operators with $ p \in L_2(0,1) $ and $ q \in W_2^{-1}(0,1) $, including singularities such as Dirac delta functions.
• To develop a constructive algorithm for recovering the potential $ p $ and its primitive $ r $ from the spectra of the problem under Dirichlet and mixed boundary conditions.
• To extend the inverse spectral theory for quadratic operator pencils by linking them to Dirac operators with $ h $-shifted AKNS-type potentials.
• To provide a framework applicable to broader classes of inverse problems involving spectral data such as spectra and norming constants or Hochstadt–Lieberman-type mixed data.

Proposed method

• Reduce the energy-dependent Sturm-Liouville problem to a quadratic operator pencil and associate it with a Dirac operator of a special form via spectral transformation techniques.
• Construct a transformation operator that relates the Dirac operator with potential $ Q $ in $ h $-shifted AKNS form to a canonical Dirac operator, preserving spectral data.
• Use the inverse spectral theory of Dirac operators to reconstruct the potential $ Q $ from the given spectra $ \bm{\mu} $ and $ \bm{\lambda} \cup \{\mu_*\} $, where $ \mu_* = (\mu_0 + \mu_1)/2 $.
• Map the reconstructed Dirac potential $ Q $ back to the original Sturm-Liouville parameters $ p $ and $ r $ using the relations $ p = (p_{22} + p_{11})/2 $, $ r = -p_{12} - \int_x^1 (p_{12}^2 - p_{11}p_{22}) \, dx $.
• Establish isospectrality between Dirac operators with potential $ P $ and $ Q $, and use the uniqueness of the isospectral set $ \operatorname{Iso}(Q) $ to recover $ P $ uniquely.
• Leverage the almost interlacing property of the spectra to ensure the hyperbolicity of the pencil, which validates the assumptions required for the inverse method.

Experimental results

Research questions

• RQ1 Can energy-dependent Sturm-Liouville operators with minimal smoothness assumptions on $ p $ and $ q $ be uniquely reconstructed from two spectra?
• RQ2 How can the inverse spectral problem for such operators be reduced to an equivalent problem for Dirac operators with special potentials?
• RQ3 What spectral conditions ensure the existence and uniqueness of the reconstruction in the presence of distributional potentials and singularities?
• RQ4 Is there a constructive algorithm to recover the potential $ p $ and its primitive $ r $ from two given spectra under Dirichlet and mixed boundary conditions?
• RQ5 Can the method be extended to other inverse problems involving different spectral data, such as spectra and norming constants?

Key findings

• The inverse spectral problem for energy-dependent Sturm-Liouville equations with $ p \in L_2(0,1) $ and $ q \in W_2^{-1}(0,1) $ is uniquely solvable from two spectra: Dirichlet and mixed boundary conditions.
• A complete reconstruction algorithm is constructed by transforming the problem into an inverse spectral problem for Dirac operators with $ h $-shifted AKNS-type potentials.
• The existence of a solution is established by constructing a potential $ Q \in \mathcal{Q}_h $ such that the spectra $ \bm{\mu} $ and $ \bm{\lambda} \cup \{\mu_*\} $ are realized by the corresponding Dirac operators.
• Uniqueness follows from the uniqueness of the isospectral set $ \operatorname{Iso}(Q) $ and the fact that $ P \in \mathcal{P}_{\mu_*} \cap \operatorname{Iso}(Q) $ is uniquely determined by the spectral data.
• The method applies to potentials with singularities such as Dirac delta functions and Coulomb-like singularities, extending previous results to broader classes of distributions.
• The reconstruction process is algorithmic: given two interlacing spectra, one can compute $ p $ and $ r $ via explicit formulas derived from the Dirac potential $ P $.