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Lower Bounds on Anderson-Localised Eigenfunctions on a Strip
1School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS UK.
Communications in Mathematical Physics
|May 9, 2022
Summary
Eigenfunctions of random Schrödinger operators decay exponentially. This study shows decay rates are strictly slower than the fastest Lyapunov exponent, matching the slowest for specific subsequences.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Spectral Theory
Background:
- Eigenfunctions of random Schrödinger operators on a strip exhibit exponential decay.
- The rate of decay is bounded below by the slowest Lyapunov exponent.
- Heuristic arguments suggest no eigenfunction decays faster than this rate.
Purpose of the Study:
- To investigate the precise rate of exponential decay for eigenfunctions of random Schrödinger operators.
- To provide analytical evidence for or against the conjecture on maximal decay rates.
Main Methods:
- Analysis of eigenfunctions' exponential decay rates.
- Consideration of subsequences for decay rate analysis.
- Focus on cases with a regular potential distribution.
Main Results:
- Demonstrated that for each eigenfunction, the decay rate along any subsequence is strictly slower than the fastest Lyapunov exponent.
- Proved the existence of a subsequence where the decay rate equals the slowest Lyapunov exponent.
Conclusions:
- The findings represent a step towards confirming the conjecture on the maximal decay rate of eigenfunctions.
- The study refines our understanding of eigenfunction localization in random media.
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