[Paper Review] Recovering Differential Operators with Nonseparated Boundary Conditions in the Central Symmetric Case

Vjacheslav Yurko|arXiv (Cornell University)|Jan 13, 2016

Spectral Theory in Mathematical Physics7 references3 citations

TL;DR

This paper solves inverse spectral problems for Sturm-Liouville operators on [0, π] with non-separated boundary conditions under central symmetry, where the potential q(x) satisfies q(x) = q(π−x). It provides necessary and sufficient conditions for solvability and constructs algorithms to recover q, a, and b from spectral data, including eigenvalues and an auxiliary sequence η. The key contribution is a complete characterization of the spectrum and uniqueness of reconstruction in the symmetric case, extending prior results to non-separated conditions with improved verifiability.

ABSTRACT

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

Motivation & Objective

• To address inverse spectral problems for Sturm-Liouville operators with non-separated boundary conditions, which are more complex than separated cases.
• To study the central symmetric case where the potential satisfies q(x) = q(π−x), reducing the required spectral information.
• To provide necessary and sufficient conditions for the solvability of inverse problems under such symmetry.
• To construct explicit reconstruction algorithms for the potential and boundary parameters from spectral data.
• To generalize existing results on periodic and Dirichlet-type problems to non-separated boundary conditions under symmetry.

Proposed method

• Uses fundamental solutions C(x,λ), S(x,λ), and ψ(x,λ) of the Sturm-Liouville equation y'' + q(x)y = λy.
• Defines characteristic functions Δ(λ), p(λ), and p⁺(λ) to relate spectral data to the potential via entire functions of order 1/2.
• Introduces the η-sequence to encode information about the boundary conditions, particularly for non-separated cases.
• Applies spectral data (eigenvalues μₙ and ηₙ) to reconstruct the potential q(x) ∈ L₂′(0,π) and parameters a, b.
• Uses the Wronskian structure and asymptotic behavior of eigenvalues to derive necessary and sufficient conditions for solvability.
• Employs conformal mapping and spectral gap analysis to verify the completeness and uniqueness of the reconstruction.

Experimental results

Research questions

• RQ1 Can the inverse spectral problem for Sturm-Liouville operators with non-separated boundary conditions be solved under central symmetry?
• RQ2 What spectral data are sufficient to uniquely reconstruct the potential and boundary parameters in the symmetric case?
• RQ3 How do the asymptotic properties of eigenvalues and gaps constrain the solvability of the inverse problem?
• RQ4 What role does the η-sequence play in characterizing the spectrum and ensuring uniqueness?
• RQ5 Can necessary and sufficient conditions for solvability be formulated in a verifiable form for non-separated conditions?

Key findings

• The eigenvalues μₙ of the non-separated BVP satisfy μₙ = n² + π⁻¹(h + (−1)ⁿ⁺¹4b) + κₙ with {κₙ} ∈ l₂, where h = 4a + ∫₀^π q(t)dt.
• The characteristic function r(λ) for the spectrum is uniquely determined by the eigenvalues via r(λ) = π(λ−μ₀)∏ₙ=1^∞ (μₙ−λ)/n².
• The η-sequence, defined by ηₙ = sign(|θ′(π,νₙ)| − |b|), is in J′ and satisfies ηₙ = 1 for all n > N, ensuring stability and uniqueness.
• Necessary and sufficient conditions for solvability of the inverse problem are given by the asymptotic behavior (26) and the maximum condition (28) on |r(λ)| over spectral gaps.
• Uniqueness of the potential q(x) ∈ L₂′(0,π) and parameters a, b is established from the spectral data μₙ and ηₙ.
• The method allows full reconstruction of the potential and boundary parameters using only the spectrum and the η-sequence, reducing data requirements in the symmetric case.