A useful formula for periodic Jacobi matrices on trees

Area of Science:

• Spectral theory
• Mathematical physics
• Graph theory

Background:

• Periodic Jacobi matrices are fundamental in spectral theory.
• Understanding the density of states is crucial for analyzing spectral properties.
• Existing methods for spectral analysis on trees are limited.

Purpose of the Study:

• Introduce a novel function for the density of states for periodic Jacobi matrices on trees.
• Develop a formula for this function using resolvent entries and half-tree restrictions.
• Streamline proofs of established theorems in spectral theory.

Main Methods:

• Definition of a new density of states function.
• Derivation of a formula relating the function to matrix resolvent entries.
• Application of the formula to prove gap labeling and Aomoto index theorems.
• Extension of the formula to the Anderson model on trees.

Main Results:

• A new formula for the density of states function for periodic Jacobi matrices on trees.
• The formula is analogous to the one-dimensional Thouless formula.
• Streamlined and complete proofs for the gap labeling theorem.
• A sketched proof for the Aomoto index theorem.
• A version of the formula for the Anderson model on trees.

Conclusions:

• The introduced formula provides a powerful tool for spectral analysis on trees.
• This work simplifies and unifies proofs of important spectral theorems.
• The results have implications for understanding disordered systems (Anderson model) on tree structures.