Simplicity of Eigenvalues in the Anderson Model

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 2(Z), where ∆(x, y) = 1 if |x − y| = 1 and zero otherwise, and the random potential Vω = {Vω(x), x ∈ Zd} 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 ∈ 2(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〉 := √ 1 + |x|2. Theorem. The Anderson Hamiltonian Hω cannot have an eigenvalue with two linearly independent fast decaying eigenfunctions with probability one. We have the immediate corollary: Corollary. Let I be an interval of exponential localization for the Anderson Hamiltonian Hω. Then, with probability one, every eigenvalue of Hω in I is simple. This corollary was originally proved by Simon [S], who proved that in intervals of localization the vectors δx, x ∈ Zd, are cyclic for Hω with probability one, and hence the pure point spectrum is simple. Simon’s cyclicity result cannot be extended to Anderson-type Hamiltonians in the continuum. In contrast, our proof is quite transparent and intuitive. The only step in the proof that cannot presently be done in the continuum is the use of Minami’s estimate [M], stated below in (7). But some form of Minami’s estimate must hold in the continuum. When Minami’s estimate is extended to the continuum, our proof will give the simplicity of eigenvalues also for continuous Anderson-type Hamiltonians. While the simplicity of eigenvalues for Anderson-type Hamiltonians in the continuum is not presently known, Germinet and Klein [GK] have recently A.K. was supported in part by NSF Grant DMS-0200710.