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On Point Spectrum of Jacobi Matrices Generated by Iterations of Quadratic Polynomials
Benjamin Eichinger1, Milivoje Lukić2,3, Peter Yuditskii4
1School of Mathematical Sciences, Lancaster University, Lancaster, LA1 4YF UK.
Communications in Mathematical Physics
|July 3, 2026
Summary
This study shows that limit-periodic Jacobi matrices, related to polynomial Julia sets, have empty point spectra across their entire hull. This finding advances inverse spectral theory for ergodic operators with zero measure spectra.
Area of Science:
- Mathematical Physics
- Spectral Theory
- Dynamical Systems
Background:
- Point spectra of Jacobi matrices can vary within their hull.
- Ergodic operators with zero measure spectra are key in mathematical physics.
- The Julia set of z^2 - lambda is a model for studying spectral properties.
Purpose of the Study:
- To investigate the spectral properties of limit-periodic Jacobi matrices associated with a specific polynomial Julia set.
- To determine if the point spectrum is empty for all elements within the hull of these matrices.
- To contribute to the inverse spectral theory of ergodic operators.
Main Methods:
- Analysis of the hull of the limit-periodic Jacobi matrix.
- Utilizing the equilibrium measure of the Julia set for the polynomial z^2 - lambda.
- Introduction and application of a matrix version of Ruelle operators.
Main Results:
- Proved that every element of the studied hull has an empty point spectrum.
- Demonstrated that the point spectrum does not depend on the element of the hull for this specific case.
- Established a connection between Ruelle operators and spectral properties of Jacobi matrices.
Conclusions:
- The hull of the limit-periodic Jacobi matrix for z^2 - lambda (large lambda) has an empty point spectrum.
- This result has implications for understanding inverse spectral theory and ergodic operators.
- Matrix Ruelle operators provide a novel tool for spectral analysis.
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