The Method of Gauge Transform - Periodic and Ergodic Spectral Problems

Explore the method of Gauge transform in this lecture by Jeffrey Galkowski, part of the SMS 2023 series on Periodic and Ergodic Spectral Problems. Delve into the construction of the Gauge transform, examining crucial aspects such as cutoff, top order, unitary invariance, and Fourier multipliers. Investigate resonance zones, finite dimensional decomposition, and unitary transformation. Analyze the monotonicity, unique solutions, and sum of eigenvalues. Gain insights into analyticity, determinants, and derivatives as they relate to the Gauge transform method.

Introduction
What do we want
Gauge transform construction
Cutoff
Top order
Unitarily invariant
Fourier multiplier
Resonance zone
Finite dimensional
Decomposition
unitary transformation
monotoning
crucial fact
unique solution
sum of eigenvalues
eigenvalues
analyticity
determinant
derivative
check