[Paper Review] An inverse spectral problem for second-order functional-differential pencils with two delays

Sergey Buterin, M.A. Malyugina+1 more|arXiv (Cornell University)|Oct 27, 2020

Spectral Theory in Mathematical Physics17 references4 citations

TL;DR

This paper solves an inverse spectral problem for a second-order functional-differential pencil with two constant delays, proving uniqueness and providing a constructive algorithm to recover the coefficients from two spectra sharing one boundary condition. The key contribution is a necessary and sufficient condition for solvability and a complete reconstruction procedure, resolving long-standing open questions in the field of operators with delay.

ABSTRACT

We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary condition. The uniqueness theorem is proved and a constructive procedure for solving this inverse problem along with necessary and sufficient conditions for its solvability is obtained. Moreover, we give a survey on the contemporary state of the inverse spectral theory for operators with delay. The pencil under consideration generalizes Sturm-Liouville-type operators with delay, which allows us to illustrate essential results in this direction, including recently solved open questions.

Motivation & Objective

• To address the inverse spectral problem for a second-order functional-differential pencil with two constant delays, generalizing Sturm–Liouville operators with delay.
• To determine the coefficients of the differential pencil from the spectra of two boundary value problems sharing one common boundary condition.
• To establish necessary and sufficient conditions for the solvability of the inverse problem and provide a constructive reconstruction procedure.
• To resolve open questions in inverse spectral theory for operators with delay, particularly regarding uniqueness and identifiability of potentials.

Proposed method

• Formulates the functional-differential pencil as a Sturm–Liouville-type operator with two constant delays, incorporating a spectral parameter in the differential equation.
• Analyzes the spectral data by deriving asymptotic expansions of the characteristic functions associated with the boundary value problems.
• Derives integral representations for the spectral data using transformed functions and introduces auxiliary functions to relate the spectra to the coefficients.
• Constructs a system of equations involving the spectra and the coefficients, enabling the recovery of the potential functions through spectral data.
• Applies a transformation method to reduce the problem to a Regge-type problem, allowing the use of characteristic function representations.
• Uses the spectral data to reconstruct the coefficients via a system of integral equations and proves that the solution is unique under the derived conditions.

Experimental results

Research questions

• RQ1 Can the coefficients of a second-order functional-differential pencil with two constant delays be uniquely reconstructed from the spectra of two boundary value problems sharing one common boundary condition?
• RQ2 What are the necessary and sufficient conditions for the solvability of this inverse spectral problem?
• RQ3 Does the uniqueness result for the potential function remain valid when the delay is less than π/2, particularly in the range (0, π/2)?
• RQ4 Can the spectra of two boundary value problems uniquely determine both potential functions when there are two distinct delays?
• RQ5 Are there cases where different potential functions produce the same spectra, indicating non-uniqueness in the inverse problem?

Key findings

• The paper proves a uniqueness theorem for the inverse spectral problem of recovering the coefficients of a second-order functional-differential pencil with two constant delays from two spectra sharing one boundary condition.
• A constructive procedure for solving the inverse problem is provided, based on spectral data and integral representations of the characteristic functions.
• Necessary and sufficient conditions for the solvability of the inverse problem are derived, ensuring that the reconstruction is possible if and only if these conditions are satisfied.
• The paper resolves an open question by showing that for delays in (0, π/2), the potential cannot be uniquely determined from two spectra alone, as demonstrated by a counterexample with a one-parametric family of iso-bispectral potentials.
• The counterexample shows that even for distinct delays, the spectra of two boundary value problems may not uniquely determine the individual potential functions, refuting earlier claims in [19] and [20].
• The paper establishes that the uniqueness subdomain for the potential function, previously thought to be maximal, is indeed unimprovable, as shown by the existence of a one-parametric family of different potentials yielding the same spectra.