Simplicity of Eigenvalues in the Anderson Model Abel Klein and Stanislav Molchanov

We give a simple, transparent, and intuitive proof that all eigenvalues of the Anderson model in the region of localization are simple. The Anderson tight binding model is given by the random Hamiltonian Hω = −∆ + Vω on l 2(Z), where ∆(x, y) = 1 if |x − y| = 1 and zero otherwise, and the random potential Vω = {Vω(x), x ∈ Z } consists of independent identically distributed random variables whose common probability distribution μ has a bounded density ρ. It is known to exhibit exponential localization at either high disorder or low energy [FMSS, DK, AM]. We prove a general result about eigenvalues of the Anderson Hamiltonian with fast decaying eigenfunctions, from which we conclude that in the region of exponential localization all eigenvalues are simple. We call φ ∈ l2(Zd) fast decaying if it has β-decay for some β > 5d 2 , that is, |φ(x)| ≤ Cφ〈x〉 −β for some Cφ < ∞, where 〈x〉 := √