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Axis and low-dimensional cases of Levitin's disk-intersection conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and notation.
An indefinite matrix pencil $A-\lambda B$ has eigenvalues where its determinant vanishes. A Nevanlinna function maps the upper half-plane to itself and the lower half-plane to itself under conjugate symmetry; meromorphic examples have real poles. A Schauder basis permits a unique norm-convergent ordered expansion. Counts in the source are zeros and poles, with a multiplicity convention requiring specification for a proof.
Catalogue wording retained for traceability.
The record bundles primary items 11.1--13.1.
  1. Let $n\ge1$, $0<c<2$, and let $A$ be the $2n\times2n$ tridiagonal matrix with diagonal entries $c$ and adjacent off-diagonal entries $1$. Let $B=\operatorname{diag}(I_n,-I_n)$. Prove that every non-real $\lambda\in\sigma(BA)$ satisfies both $|\lambda-c|\le2$ and $|\lambda+c|\le2$.
  2. Extend the spectral results for $(-d^2/dx^2+c/(1+|x|))\psi=\lambda\operatorname{sgn}(x)\psi$ to more general potential functions, specifying the admissible class and conclusions.
  3. Let $m_j$ be meromorphic Nevanlinna functions and $I\subset\mathbb R$ a nonempty interval such that the number of poles in every nonempty subinterval $J\subset I$ tends to infinity with $j$. If $g$ is analytic on a complex neighborhood of $I$, show for every open complex neighborhood $U$ of $J$ that the number of zeros of $m_j-g$ in $U$ tends to infinity; also show that the difference between the numbers of such zeros and poles of $m_j$ in $U$ is bounded by a constant independent of $j$. The source records the case where $\lim_jj^{-1}\#\{\text{poles in }J\}$ exists.
  4. For $V(x)=\sum_{k=-m}^m v_ke^{2ikx}$, $v_k\in\mathbb C$, classify coefficient sets for which eigenfunctions of $-d^2/dx^2+V$ on $(0,\pi)$ with periodic boundary conditions form a Schauder basis of $L^2(0,\pi)$. The source records the case $V=e^{-2ix}+be^{2ix}$ as affirmative exactly when $|b|=1$.
Scope and source qualification.
The original broad Sturm–Liouville extension and the zero/pole count conventions are preserved as questions to specify. Spectral domains and generalized eigenfunctions must be treated where needed; a statement about the named matrix alone does not settle this multi-question record. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement. The statement covers the original source questions rather than only the title’s subcase.

Short English statement

Resolve an indefinite-pencil enclosure, generalize an indefinite Sturm–Liouville model, compare Nevanlinna zeros and poles, and classify periodic Schrödinger bases.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-130-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-130

Axis and low-dimensional cases of Levitin's disk-intersection conjecture

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and notation.
An indefinite matrix pencil $A-\lambda B$ has eigenvalues where its determinant vanishes. A Nevanlinna function maps the upper half-plane to itself and the lower half-plane to itself under conjugate symmetry; meromorphic examples have real poles. A Schauder basis permits a unique norm-convergent ordered expansion. Counts in the source are zeros and poles, with a multiplicity convention requiring specification for a proof.
Catalogue wording retained for traceability.
The record bundles primary items 11.1--13.1.
  1. Let $n\ge1$, $0<c<2$, and let $A$ be the $2n\times2n$ tridiagonal matrix with diagonal entries $c$ and adjacent off-diagonal entries $1$. Let $B=\operatorname{diag}(I_n,-I_n)$. Prove that every non-real $\lambda\in\sigma(BA)$ satisfies both $|\lambda-c|\le2$ and $|\lambda+c|\le2$.
  2. Extend the spectral results for $(-d^2/dx^2+c/(1+|x|))\psi=\lambda\operatorname{sgn}(x)\psi$ to more general potential functions, specifying the admissible class and conclusions.
  3. Let $m_j$ be meromorphic Nevanlinna functions and $I\subset\mathbb R$ a nonempty interval such that the number of poles in every nonempty subinterval $J\subset I$ tends to infinity with $j$. If $g$ is analytic on a complex neighborhood of $I$, show for every open complex neighborhood $U$ of $J$ that the number of zeros of $m_j-g$ in $U$ tends to infinity; also show that the difference between the numbers of such zeros and poles of $m_j$ in $U$ is bounded by a constant independent of $j$. The source records the case where $\lim_jj^{-1}\#\{\text{poles in }J\}$ exists.
  4. For $V(x)=\sum_{k=-m}^m v_ke^{2ikx}$, $v_k\in\mathbb C$, classify coefficient sets for which eigenfunctions of $-d^2/dx^2+V$ on $(0,\pi)$ with periodic boundary conditions form a Schauder basis of $L^2(0,\pi)$. The source records the case $V=e^{-2ix}+be^{2ix}$ as affirmative exactly when $|b|=1$.
Scope and source qualification.
The original broad Sturm–Liouville extension and the zero/pole count conventions are preserved as questions to specify. Spectral domains and generalized eigenfunctions must be treated where needed; a statement about the named matrix alone does not settle this multi-question record. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement. The statement covers the original source questions rather than only the title’s subcase.

Short English statement

Resolve an indefinite-pencil enclosure, generalize an indefinite Sturm–Liouville model, compare Nevanlinna zeros and poles, and classify periodic Schrödinger bases.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Verified both disk enclosures for n=1 and proved neutral-block identities for arbitrary nonreal pencil eigenvectors. All-n enclosure and the three additional spectral questions remain unresolved.
Sources and literature check.
Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.
The smallest indefinite pencil.
For $n=1$, the matrix is $BA=\begin{pmatrix}c&1\\-1&-c\end{pmatrix}$, whose characteristic polynomial is $\lambda^2+1-c^2$. Nonreal eigenvalues exist exactly for $0<c<1$ and are $\lambda=\pm i\sqrt{1-c^2}$. Both satisfy $|\lambda-c|=|\lambda+c|=1$, strictly within the proposed disks of radius two. For $c=1$ the double eigenvalue is real zero, and for $1<c<2$ both eigenvalues are real, so the nonreal assertion is vacuous.
A general neutral-vector identity.
For any $n$, $BA$ is self-adjoint for the indefinite form with matrix $B$: $(BA)^*B=B(BA)=A$. If $BAu=\lambda u$ with $\operatorname{Im}\lambda\ne0$, comparison with its conjugate gives $(\lambda-\bar\lambda)u^*Bu=0$, hence the squared Euclidean norms of the first and second blocks of $u$ agree. Also $u^*Au=\lambda u^*Bu=0$. These identities exclude a proof based on a positive definite Rayleigh quotient and identify the role of the coupling across the block boundary; by themselves they do not yield both disk inequalities.
Other questions and first gap.
A bound exploiting that single boundary coupling strongly enough for all $n$ remains missing. The broader indefinite Sturm--Liouville extension, Nevanlinna zero/pole counts and periodic Schrödinger Schauder-basis classification are not consequences of this matrix calculation and remain unresolved. Multiplicity conventions and operator domains in those subproblems have not been supplied ad hoc. The known elementary $n=1$ check and neutrality identity are not counted as a solution of the multi-question record.
Verification and adversarial audit.
Checked characteristic polynomial, c=1 and c>1 boundary cases, indefinite adjoint identity and both norm-neutrality consequences. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked characteristic polynomial, c=1 and c>1 boundary cases, indefinite adjoint identity and both norm-neutrality consequences. Scope remains limited: Use the block coupling to derive both radius-two disk bounds for arbitrary n; separate analytic arguments are required for the other source items.

Coordinator checked the two-dimensional characteristic polynomial, disk distances and indefinite-form neutral-vector identities. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Use the block coupling to derive both radius-two disk bounds for arbitrary n; separate analytic arguments are required for the other source items.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

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