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A one-coupling J-Schur certificate for projected spectral enclosures

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.
For compact $K$, its approximation number is $\alpha_N(K)=\inf_{\operatorname{rank}F<N}\|K-F\|$. Write $\operatorname{spr}(A)=\sup\{|z|:z\in\sigma(A)\}$. Resonances are poles of a suitably continued resolvent; an isoresonant family has the same resonance data. The source uses a J-self-adjoint structure and refers to J as a conjugation; its exact convention must be preserved in a chosen approximation scheme.
Catalogue wording retained for traceability.
The record bundles source items 17.1--19.1.
  1. Devise projected-space methods for rough rigorous and sharp useful enclosures of spectra of J-self-adjoint operators, incorporating the source’s J structure, and assess consequences for their evolution problems.
  2. For Schrödinger operators $-\Delta+V$ on $\mathbb R^d$ with bounded compactly supported potentials, find spectral or pseudospectral data that distinguish members of isoresonant complex-potential families in dimensions $d\ge2$, including nontrivial smooth complex potentials with no resonances. Ask the analogous question for isospectral families in settings such as the circle.
  3. For a Banach-space operator $A$ and compact perturbation $K$ with $\alpha_N(K)\to0$, estimate discrete eigenvalue counts of $A+K$ outside disks using resolvent growth near $\sigma(A)$. In particular classify operators whose resolvent has polynomial growth toward the spectrum; classify those admitting $M\ge1$ such that $\|(\lambda-A)^{-1}\|\le M/\operatorname{dist}(\lambda,\sigma(A))$ for every resolvent parameter; and classify those admitting $\|(\lambda-A)^{-1}\|\le M/(|\lambda|-\operatorname{spr}(A))$ whenever $|\lambda|>\operatorname{spr}(A)$.
Scope and source qualification.
The long counting bound in the PDF motivates the question rather than constituting an additional conjecture. All three requested classifications and both isoresonant settings are retained. J is not silently changed from conjugation to a linear indefinite-metric involution. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement.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

Develop reliable projected enclosures, distinguish isoresonant potentials, and classify operators with controlled resolvents.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-132-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-132

A one-coupling J-Schur certificate for projected spectral enclosures

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.
For compact $K$, its approximation number is $\alpha_N(K)=\inf_{\operatorname{rank}F<N}\|K-F\|$. Write $\operatorname{spr}(A)=\sup\{|z|:z\in\sigma(A)\}$. Resonances are poles of a suitably continued resolvent; an isoresonant family has the same resonance data. The source uses a J-self-adjoint structure and refers to J as a conjugation; its exact convention must be preserved in a chosen approximation scheme.
Catalogue wording retained for traceability.
The record bundles source items 17.1--19.1.
  1. Devise projected-space methods for rough rigorous and sharp useful enclosures of spectra of J-self-adjoint operators, incorporating the source’s J structure, and assess consequences for their evolution problems.
  2. For Schrödinger operators $-\Delta+V$ on $\mathbb R^d$ with bounded compactly supported potentials, find spectral or pseudospectral data that distinguish members of isoresonant complex-potential families in dimensions $d\ge2$, including nontrivial smooth complex potentials with no resonances. Ask the analogous question for isospectral families in settings such as the circle.
  3. For a Banach-space operator $A$ and compact perturbation $K$ with $\alpha_N(K)\to0$, estimate discrete eigenvalue counts of $A+K$ outside disks using resolvent growth near $\sigma(A)$. In particular classify operators whose resolvent has polynomial growth toward the spectrum; classify those admitting $M\ge1$ such that $\|(\lambda-A)^{-1}\|\le M/\operatorname{dist}(\lambda,\sigma(A))$ for every resolvent parameter; and classify those admitting $\|(\lambda-A)^{-1}\|\le M/(|\lambda|-\operatorname{spr}(A))$ whenever $|\lambda|>\operatorname{spr}(A)$.
Scope and source qualification.
The long counting bound in the PDF motivates the question rather than constituting an additional conjecture. All three requested classifications and both isoresonant settings are retained. J is not silently changed from conjugation to a linear indefinite-metric involution. 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

Develop reliable projected enclosures, distinguish isoresonant potentials, and classify operators with controlled resolvents.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived a Schur-complement exclusion certificate and classified finite matrices with reciprocal-distance resolvent growth. No J-compatible infinite-dimensional scheme, isoresonance discriminator or full resolvent/counting classification.
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.
A projected-space exclusion certificate.
Let a bounded operator on an orthogonal sum be $T=\begin{pmatrix}A&B\\C&D\end{pmatrix}$, with the first block finite dimensional. If $D-z$ is invertible, block elimination proves that $T-z$ is invertible exactly when $S(z)=A-z-B(D-z)^{-1}C$ is invertible. If $A-z$ is invertible and \[\|(A-z)^{-1}\|\,\|B\|\,\|(D-z)^{-1}\|\,\|C\|<1, \] then factoring $S=(A-z)[I-(A-z)^{-1}B(D-z)^{-1}C]$ and summing a Neumann series certifies exclusion of $z$ from the spectrum. A finite compression by itself omits the decisive complement resolvent.
A finite-dimensional classification test.
For a complex matrix, a bound $\|(T-z)^{-1}\|\le M/\operatorname{dist}(z,\sigma(T))$ for every resolvent parameter holds if and only if the matrix is diagonalizable. Sufficiency follows from $T=S\Lambda S^{-1}$, with $M=\|S\|\|S^{-1}\|$. For necessity, a Jordan block of size $r\ge2$ has resolvent term proportional to $(z-\lambda)^{-r}$; applying it to the last basis vector gives a norm bounded below by a positive constant times that power near $\lambda$, even after fixed invertible similarity. This contradicts a reciprocal-distance bound.
Gap and conventions.
Neither argument uses or changes the source's antilinear conjugation $J$. A useful $J$-compatible projection scheme must establish complement estimates respecting the actual structure. Infinite-dimensional polynomial resolvent classifications, compact-perturbation counting, and data distinguishing isoresonant or isospectral potentials do not follow from the finite-matrix test. All remain unresolved, including the exterior-disk classification, which is different from distance-to-spectrum control.
Verification and adversarial audit.
Checked block factorization order, strict Neumann inequality, Jordan-pole growth, similarity constants and the antilinear J convention. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked block factorization order, strict Neumann inequality, Jordan-pole growth, similarity constants and the antilinear J convention. Scope remains limited: Control the infinite-dimensional complement resolvent while preserving the source conjugation structure, then address the separate classification and resonance questions.

Coordinator checked the Schur-complement exclusion criterion and finite-dimensional Jordan-resolvent classification, with infinite-dimensional scope excluded. 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. Control the infinite-dimensional complement resolvent while preserving the source conjugation structure, then address the separate classification and resonance questions.

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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