“The eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials.”

“For every irrational ω with β(ω) < L, where L > 0 denotes the Lyapunov exponent, and every non-resonant phase θ, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions.”

“All models are exactly-solvable, admit algebraic forms for the Hamiltonian and integrals, have polynomial eigenfunctions, hidden algebraic structure, and possess a polynomial algebra of integrals.”

“Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models.”

“The main contribution is the derivation of unified gradient estimates that connect the near-boundary and far-field regions through a characteristic length scale determined by the Robin parameter, yielding a global description of how boundary effects penetrate into the exterior domain.”

“The provided context is minimal, only indicating the source as ArXiv.”