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Phys-33 · physics · mathematics · Physics

Non-self-adjoint spectral theory: completeness, reconstruction, and matrix-function bounds

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
This collection asks the following questions about non-self-adjoint spectral theory.
  1. For $A_\alpha=-d^2/dx^2+i|x|^\alpha$ on the real line, or on the positive half-line with a Dirichlet condition at zero, do the eigenfunctions form a complete system when $0<\alpha\leq2/3$?
  2. On $\mathbb R^2_+=\{(x,y):y>0\}$, consider \[A=-\partial_x^2-(\partial_y-ix^2/2)^2+icy,\qquad D(A)=H^1_0(\mathbb R^2_+)\cap\{u:Au\in L^2(\mathbb R^2_+)\}.\] Is its spectrum nonempty for every parameter $c$?
  3. Let $f$ be continuous, complex-valued, two-periodic, odd, and symmetric about $1/2$. Put $f_n(x)=f(nx)$, and define $A$ initially by $A(\sqrt2\sin(n\pi x))=f_n$. For $p>1$, $p\neq2$, find necessary and sufficient conditions on the Fourier coefficients of $f$ for $A$ to be boundedly invertible on $L^p(0,1)$, equivalently for $(f_n)$ to be a Schauder basis.
  4. In $u'(t)=Au(t)$ with $A=L+B$, where $L$ is known self-adjoint and $B$ is unknown non-self-adjoint, recover $A$ or its discrete spectrum from observations $\omega(t)=\langle u(t),g\rangle$. How should the initial state and observation vector be chosen to detect all eigenvalues and generalized eigenvectors? What convergence and smoothness suffice for the spectral expansion? Can the complex eigenvalues and their multiplicities be recovered in finite time, with the smallest possible finite number of observations?
  5. Under what hypotheses can the instability index of a non-self-adjoint differential operator be determined from a finite matrix obtained by restriction to trigonometric polynomials? The instability index counts unstable eigenvalues with multiplicity; the question includes controlling truncation errors and the required matrix size.
  6. For a bounded operator on a Banach space, characterize when $\|A\|-\operatorname{spr}(A)$ is zero or strictly positive. Determine the best factor $m\in(0,1]$ in $\operatorname{spr}(A)\leq m\|A\|$ for the operator classes under consideration, including the relevance to discrete spectra of compact perturbations.
  7. For a nonnormal $n\times n$ matrix, set \[s(A,\varepsilon)=\inf_{V^{-1}(A+\Delta)V\ \mathrm{diagonal}}\left(\varepsilon\|V\|\|V^{-1}\|+\|\Delta\|\right).\] Does a constant $C_n$ independent of $A$ give $s(A,\varepsilon)\leq C_n\sqrt\varepsilon$?
  8. For a bounded operator $A$ with numerical range $W(A)$, is there a universal $C\geq2$ such that $\|f(A)\|\leq C\max_{z\in W(A)}|f(z)|$ for every analytic $f$ defined on a neighborhood of that range? Can $C\leq11.08$ be used?

Formulation references: [S1], [S2].

Short English statement

Resolve the listed completeness, reconstruction, stability, approximation, and numerical-range questions for non-self-adjoint operators.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-33-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-33

The constant-two numerical-range theorem addresses the Crouzeix matrix-function subquestion. Spectral completeness, inverse spectral problems, instability, and the other approximation clauses in the same entry remain separate.

Non-self-adjoint spectral theory: completeness, reconstruction, and matrix-function bounds

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 325; 2026-09-26), the asserted result is that finite matrices obey the sharp constant-two numerical-range bound for matrix-valued polynomials in every coefficient size, including degenerate numerical ranges; In manuscript 2 (family 325; 2026-09-23), the asserted result is that the sharp complete constant-two numerical-range estimate holds for bounded operators without restrictions on Hilbert-space dimension or separability, also for the specified holomorphic and rational calculus.

Mathematical statement of the claimed result

Manuscript 1.
For integers $n,m\ge1$, $d\ge0$, a complex matrix $A\in M_n(\mathbb C)$ and coefficients $B_k\in M_m(\mathbb C)$, let \[ P(z)=\sum_{k=0}^d B_kz^k,\qquad P[A]=\sum_{k=0}^d A^k\otimes B_k, \quad W(A)=\{x^*Ax:x\in\mathbb C^n,\ \|x\|_2=1\}. \] With Hilbert-space operator norms, the claimed complete inequality is \[ \|P[A]\|\le2\max_{z\in W(A)}\|P(z)\|. \] The factor $2$ is asserted sharp uniformly over all three integer parameters. Matrix-valued coefficients are allowed; the scalar polynomial case follows with $m=1$.
Manuscript 2.
For $n,m\ge1$, $A\in M_n(\mathbb C)$, $d\ge0$ and arbitrary $B_0,\ldots,B_d\in M_m(\mathbb C)$, write $P(z)=\sum_jB_jz^j$ and $W(A)=\{x^*Ax:\|x\|_2=1\}$. The claimed complete polynomial inequality is \[ \left\|\sum_{j=0}^dA^j\otimes B_j\right\|\le2\max_{z\in W(A)}\|P(z)\|. \] Hilbert-space operator norms are used, and the constant two is claimed sharp independently of matrix sizes and polynomial degree. The scalar case takes $m=1$.

Outline of the argument

For manuscript 1 (family 325; 2026-09-26), exterior conformal coordinates produce Faber coefficient estimates and a contractive correction kernel. Parseval separation of positive and negative modes controls amplified singular vectors, and enclosing-domain limits include point and segment numerical ranges.

For manuscript 2 (family 325; 2026-09-23), for finite matrices, an optimal positive metric makes the conformal disk coordinate contractive. Endpoint extremal data yield a common continuous boundary density whose ordered comparison forces condition number at most two; domain approximation and finite compressions extend the bound to general operators.

Formal verification

For manuscript 1 (family 325; 2026-09-26), OpenAI lists Lean statements for this assertion: finite matrices obey the sharp constant-two numerical-range bound for matrix-valued polynomials in every coefficient size, including degenerate numerical ranges.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 2 (family 325; 2026-09-23), OpenAI lists Lean statements for this assertion: the sharp complete constant-two numerical-range estimate holds for bounded operators without restrictions on Hilbert-space dimension or separability, also for the specified holomorphic and rational calculus.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are CompleteCrouzeix.lean, CrouzeixHilbert.lean, DirectCrouzeix.lean, HilbertCrouzeix.lean, StructuralCrouzeix.lean.

Their named declarations include OAI.CompleteCrouzeix.main, OAI.CrouzeixHilbert.hilbert, OAI.DirectCrouzeix.complete_crouzeix, OAI.DirectCrouzeix.uniform_sharpness, OAI.HilbertCrouzeix.exists_finite_compression, OAI.HilbertCrouzeix.numericalClosure_geometry, OAI.HilbertCrouzeix.rational_denominator_isUnit, OAI.HilbertCrouzeix.zeroConclusion, OAI.StructuralCrouzeixReference.challenge.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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