Absence of Eigenvalues for Pisot Substitution Operators

Definitions and mathematical statement

Let $A$ be a finite nonempty alphabet and let $\sigma:A\to A^+$ be a substitution, extended to finite words by concatenation. Its incidence matrix is $M_\sigma=(m_{ab})_{a,b\in A}$, where $m_{ab}$ counts occurrences of $a$ in $\sigma(b)$. The substitution is primitive if some positive power of $M_\sigma$ has every entry positive. Using the usual substitution-language convention, its two-sided hull is \[ \Omega=\{\omega\in A^{{\mathbb{Z}}}:\text{every finite contiguous block of }\omega\text{ occurs in some }\sigma^k(a),\ a\in A,\ k\geq0\}. \] Let $\ell^2({\mathbb{Z}})=\{\psi:{\mathbb{Z}}\to{\mathbb{C}}:\sum_{n\in{\mathbb{Z}}}|\psi(n)|^2<\infty\}$. For an arbitrary injective map $f:A\to{\mathbb{R}}$ and each $\omega\in\Omega$, define the bounded self-adjoint operator $H_{\omega,f}$ on all of $\ell^2({\mathbb{Z}})$ by \[ (H_{\omega,f}\psi)(n)=\psi(n+1)+\psi(n-1)+f(\omega_n)\psi(n),\qquad n\in{\mathbb{Z}}. \] These are operators on the whole integer line; no finite-endpoint boundary conditions are imposed. Define \[ \Omega_c(f)=\{\omega\in\Omega:H_{\omega,f}\text{ has no eigenvalues}\}. \] Here absence of eigenvalues means \[ \forall E\in{\mathbb{R}}\quad\forall\psi\in\ell^2({\mathbb{Z}}),\qquad H_{\omega,f}\psi=E\psi\ \Longrightarrow\ \psi=0. \] A Pisot number is a real algebraic integer greater than $1$ whose other algebraic conjugates all have modulus less than $1$. However, the supplied record does not define the phrase "Pisot substitution." Requiring the Perron--Frobenius eigenvalue of $M_\sigma$ to be a Pisot number does not specify the same condition as requiring every other eigenvalue of $M_\sigma$ to have modulus less than $1$; additional irreducibility conventions may also matter. No convention is selected here.

Question, subject to this unresolved qualification. If $\sigma$ is a primitive Pisot substitution in the intended source sense, is it true that, for every injective $f:A\to{\mathbb{R}}$, \[ \Omega_c(f)=\Omega? \] Equivalently, must every $\omega\in\Omega$, every $E\in{\mathbb{R}}$, and every $\psi\in\ell^2({\mathbb{Z}})$ satisfy \[ \left[\ \psi(n+1)+\psi(n-1)+f(\omega_n)\psi(n)=E\psi(n)\text{ for all }n\in{\mathbb{Z}}\ \right]\ \Longrightarrow\ \psi=0? \] The supplied statement includes no explicit aperiodicity, irreducibility, or unimodularity hypothesis. Whether any such restriction belongs to the intended meaning of "Pisot" requires source clarification; none is silently added.

Short English statement

For a primitive substitution described by the source as Pisot, assign distinct real potential values to its letters. Must every resulting whole-line Schrödinger operator, for every sequence in the substitution hull, have no nonzero square-summable eigenfunction? The intended definition of a Pisot substitution remains unresolved.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001212$, source problem code catalogue entry, title "Absence of Eigenvalues for Pisot Substitution Operators," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2009 and supplies no individual proposer attribution.
• Supplied report citation: Mini-Workshop: The Pisot Conjecture --- From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets (2009), Oberwolfach Reports, pp. 725--766, DOI 10.4171/owr/2009/13. https://doi.org/10.4171/owr/2009/13. This is also the supplied literature source labelled "original OWR report."
• Supplied literature source labelled "primitive-substitution Schrödinger spectrum paper": https://arxiv.org/abs/1311.0954. No authors, theorem numbers, or precise supporting passages are supplied in the record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the assertion as a relative Pisot conjecture, and reports that only special-substitution results and measure or dense-$G_\delta$ results were known, rather than the full assertion. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its accompanying verification note describes balanced mathematical delimiters for substitution, hull, operator, function-space, and spectral-set notation. That supplied check does not establish mathematical correctness or resolve the meaning of "Pisot substitution."