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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 $𝐓=$ $ℝ/\left(2\pi ℤ\right)$. Let $Q:{𝐓}^{d}\to ℝ$ be a real analytic function. The Schrödinger operator on $L\left(1,\infty \right)$ is given by

${H}_{0}=-\frac{{d}^{2}}{d{x}^{2}}+{q}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}{q}_{0}\left(x\right)=Q\left(\omega x\right),$

where $\omega \in {𝐓}^{d}$ is fixed. The author investigates decaying perturbations of the quasiperiodic operator ${H}_{0}$. That is, he considers the operator

${H}_{1}=-\frac{{d}^{2}}{d{x}^{2}}+{q}_{1}\left(x\right),\phantom{\rule{1.em}{0ex}}{q}_{1}\left(x\right)={q}_{0}\left(x\right)+{\Delta }q\left(x\right)$

for some function ${\Delta }q$. Assume that ${\Delta }q\left(x\right)\to 0$ as $x\to \infty$, and let $E$, $E\ge {E}_{0}$, be a boundary point of the essential spectrum of ${H}_{0}$ (${E}_{0}$ is a certain constant). Then

${\sigma }_{\text{ess}}\left({H}_{1}\right)={\sigma }_{\text{ess}}\left({H}_{0}\right)=ℝ\setminus \bigcup _{n}{G}_{n}$

for open sets ${G}_{n}$ and there exists a constant $K=K\left(E\right)$ such that $E$ is an accumulation point of eigenvalues of ${H}_{1}$ if

$\underset{x\to \infty }{lim sup}K{\Delta }q\left(x\right){x}^{2}<-\frac{1}{4}$

and $E$ is not an accumulation point of eigenvalues of ${H}_{1}$

$\underset{x\to \infty }{lim inf}K{\Delta }q\left(x\right){x}^{2}>-\frac{1}{4}·$

Also, similar results are given for the operator ${H}_{\mu }^{\gamma }\equiv {H}_{0}+\frac{\mu }{{x}^{\gamma }}$, where $0<\gamma \le 2$.

##### MSC:
 34L05 General spectral theory for OD operators 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34B24 Sturm-Liouville theory 34L20 Asymptotic distribution of eigenvalues for OD operators 34L40 Particular ordinary differential operators 47E05 Ordinary differential operators
##### Keywords:
Sturm-Liouville operators; oscillation theory