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On perturbations of quasiperiodic Schrödinger operators. (English) Zbl 1214.34077

Using relative oscillation theory and the reducible result of Eliasson studied perturbations of quasiperiodic Schrödinger operators.

Let 𝐓 d be a d-dimensional torus, where 𝐓= /(2π). Let Q:𝐓 d be a real analytic function. The Schrödinger operator on L(1,) is given by

H 0 =-d 2 dx 2 +q 0 (x),q 0 (x)=Q(ωx),

where ω𝐓 d is fixed. The author investigates decaying perturbations of the quasiperiodic operator H 0 . That is, he considers the operator

H 1 =-d 2 dx 2 +q 1 (x),q 1 (x)=q 0 (x)+Δq(x)

for some function Δq. Assume that Δq(x)0 as x, and let E, EE 0 , be a boundary point of the essential spectrum of H 0 (E 0 is a certain constant). Then

σ ess (H 1 )=σ ess (H 0 )= n G n

for open sets G n and there exists a constant K=K(E) such that E is an accumulation point of eigenvalues of H 1 if

lim sup x KΔq(x)x 2 <-1 4

and E is not an accumulation point of eigenvalues of H 1

lim inf x KΔq(x)x 2 >-1 4·

Also, similar results are given for the operator H μ γ H 0 +μ x γ , where 0<γ2.

34L05General spectral theory for OD operators
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34B24Sturm-Liouville theory
34L20Asymptotic distribution of eigenvalues for OD operators
34L40Particular ordinary differential operators
47E05Ordinary differential operators
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