Kotani–Last Conjecture Overview

• Kotani–Last Conjecture is a central spectral theory claim that posits ac spectrum implies almost periodic potentials, a view now refuted by recent counterexamples.
• Recent methods employ slow deformation techniques and function-theoretic approaches to construct nonalmost periodic ergodic potentials exhibiting pure ac spectrum.
• Analytic tools such as transfer matrices, Floquet theory, and Hardy spaces have refined spectral characterization and enhanced operator classification.

The Kotani–Last Conjecture is a central assertion in spectral theory, positing that, for one-dimensional ergodic operators—such as Schrödinger, Jacobi, and Dirac operators—the existence of a nontrivial absolutely continuous (ac) spectrum necessitates that the potential (or the sequence of coefficients) be almost periodic. This conjecture—long considered plausible due to rigidity phenomena observed in almost periodic models—has now been decisively refuted in several frameworks, notably through explicit counterexamples and analyses involving nonalmost periodic, ergodic, or weakly mixing potentials exhibiting pure ac spectrum. This article surveys the formal statement, developments, analytic machinery, and ramifications of the conjecture in light of recent advances.

1. Formal Statement and Historical Context

The conjecture originated from the study of ergodic one-dimensional Schrödinger operators of the form

$$(Hu)(t) = -u''(t) + V(t)u(t),\quad u \in L^2(\mathbb{R}),$$

and discrete models such as

$$(u_{n+1}) + (u_{n-1}) + V(n)u_n,\quad u \in \ell^2(\mathbb{Z}),$$

where $V$ is an ergodic potential. The central questions were:

• Does the presence of ac spectrum imply that $V$ is (uniformly) almost periodic?
• Are generalized eigenfunctions necessarily bounded on the support of the ac spectrum?

The "expected rigidity" held that ac spectrum requires almost periodicity, reinforced by established results for periodic and quasi-periodic models. The conjecture extends to Jacobi matrices, continuum models, and even CMV and Dirac operators (Avila, 2012, Damanik et al., 2014, Davis et al., 23 Sep 2025).

2. Analytic Framework and Principles

In both discrete and continuum settings, spectral analysis leverages:

• Transfer matrices: For continuum Schrödinger operators, solutions to

$$\frac{dA[V](E, t, s)}{ds} = \begin{pmatrix} 0 & -(E+V(s)) \ 1 & 0 \end{pmatrix},\quad A[V](E, t, t) = I,$$

and for discrete,

$$A[V](E, n, n+1) = \begin{pmatrix} E-V(n) & -1 \ 1 & 0 \end{pmatrix}.$$

• Floquet theory: For periodic $V$, spectral bands are prescribed by $|A[V](E)| \leq 2$ and rotation numbers are well-defined.
• Lyapunov exponents: The vanishing of $L(E)$ is intimately tied to ac spectrum and the deterministic structure of $V$.
• Parseval-type bounds: For the ac spectrum, integrated quadratic norms of eigenfunctions are bounded, e.g.,

$$\frac{1}{2\pi} \int_\Lambda (|u(E, x, n-1)|^2 + |u(E, x, n)|^2) dE \leq 1.$$

3. Construction of Counterexamples

Recent breakthroughs have demonstrated that the conjecture fails in both continuum and discrete cases.

Slow Deformation Method (Avila, 2012):

• Initiate with a periodic $V$ (bounded eigenfunctions, well-understood ac spectrum).
• "Pad" the potential: concatenate long periodic blocks with short gaps (zero potential), adjusting block lengths via a small parameter $\delta$.
• "Twist" and "slide" phases: introduce shifts and time-dependent translations to destroy the periodic alignment.
• Iterate these deformations; each step is a small $C^k$ perturbation, but the projective limit yields a potential that is not almost periodic (even weakly mixing) with substantial ac spectrum.
• The transfer matrix can be conjugated near rotations,

$B(A)AB(A)^{-1} = R_{\Theta(A)},$

where $|A| < 2$.

Key Outcome: There exist ergodic potentials $V$ for which the ac spectrum is 'large', but $V$ is not almost periodic and eigenfunctions may be unbounded on a positive measure set in the ac spectrum (Avila, 2012).

Function-Theoretic Approach (Volberg et al., 2012, Damanik et al., 2014):

• Develops a parameterization of reflectionless operators via Hardy spaces on Widom domains.
• The "Direct Cauchy Theorem" (DCT)—an analytic condition on the resolvent domain—controls almost periodicity:
  • If DCT holds, the generalized Abel map is injective, and the operator family is almost periodic.
  • If DCT fails, the map has nontrivial fibers; the operators are nonalmost periodic though they retain purely ac spectrum.
• Explicit formula:

$$\frac{1}{2\pi i} \int_E \frac{F(x)\, dx}{x-z} = F(z),\quad z \in \Omega.$$

• These dichotomies extend to Dirac and CMV operators (Damanik et al., 2014, Davis et al., 23 Sep 2025).

4. Matrix-Valued, Jacobi, and Dirac Operators

Extensions to higher-dimensional and more general models have been developed.

Matrix-Like Jacobi Operators (Oliveira et al., 2021):

• Operator:

$$[H_\omega u]_n = D^*(T^{n-1}\omega) u_{n-1} + D(T^n\omega) u_{n+1} + V(T^n\omega) u_n,$$

with symmetric $D(\omega)$ and self-adjoint $V(\omega)$.

• The ac spectrum of multiplicity $2r$ corresponds to the essential closure of the set where exactly $2r$ Lyapunov exponents vanish,

$$\overline{\mathcal{Z}_r}^{\,\rm ess} = \sigma_{ac,2r}(H_\omega).$$

• Odd multiplicity ac spectrum is impossible; symplectic structure enforces pairings.

Dirac Operators (Davis et al., 23 Sep 2025):

• Operator:

$$\Lambda_\varphi = \begin{pmatrix} i & 0 \ 0 & -i \end{pmatrix} \frac{d}{dx} + \begin{pmatrix} 0 & \varphi \ \overline{\varphi} & 0 \end{pmatrix}.$$

• Dichotomy for reflectionless operators with Parreau–Widom spectrum:
  • If DCT holds on $\Omega = \mathbb{C} \setminus E$, all potentials in the isospectral family are almost periodic.
  • If DCT fails, none are almost periodic.
• Construction methods, adapted from Avila, produce weakly mixing flows yielding nonalmost periodic potentials with purely ac spectrum.

5. Reflectionless Operators, Abel Maps, and Character Spaces

The use of inverse spectral theory and character-automorphic Hardy spaces yields a precise parameterization for isospectral families.

• Reflectionless property: For Schrödinger, Jacobi, and Dirac operators, the reflectionless condition equates boundary values of half-line Weyl–Titchmarsh functions.
• Generalized Abel map,

$$\mathcal{A}: \mathcal{D}(E) \to \pi_1(\Omega)^* \times \mathbb{T},$$

linearizes translation flows and determines almost periodicity via injectivity.

• In non-DCT cases, the mapping is 'nonunique'—distinct divisors (potentials) map to the same character; generic minimality of translation flow then excludes almost periodicity.

6. Dynamical Spectrum, Transport, and Spectral Implications

The counterexamples overturn the conception that ac spectrum enforces strong dynamical regularity.

• One can construct ergodic operators (with weakly mixing flows) having purely ac spectrum and nonalmost periodic coefficients/potentials.
• Eigenfunctions may be unbounded even when Lyapunov exponents vanish and the ac spectrum is present (Avila, 2012).
• These results necessitate a reexamination of mechanisms underpinning ac spectra, transport properties, eigenfunction localization, and dynamical spectra in one-dimensional models.

Table: Dichotomy by Direct Cauchy Theorem (DCT) Condition

Operator Type	DCT Holds	DCT Fails
Jacobi, Schrödinger	All potentials are almost periodic	None are almost periodic
Dirac	All potentials are almost periodic	None are almost periodic

7. Broader Consequences and Future Directions

The failure of the Kotani–Last Conjecture across operator classes—Schrödinger, Jacobi, CMV, Dirac—shows that the presence of ac spectrum does not imply almost periodicity or boundedness of eigenfunctions. This suggests a need for new characterizations of ac spectral types, utilizing tools from function theory on Riemann surfaces, complex analysis, and dynamical systems. The analytic criteria such as the DCT and the precise structure of isospectral character spaces afford promising avenues for fine spectral classification and the study of transport and localization in ergodic systems.

A plausible implication is that further work will investigate the interplay between analytic conditions (e.g., Widom domains, DCT validity), dynamical properties (almost periodicity, weak mixing), and spectral type in more general operator families. The versatility of projective limit constructions and character-automorphic Hardy spaces may also see expanded roles in inverse spectral theory and mathematical physics.