JournalsjstVol. 3, No. 1pp. 1–45

An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size nn contained in an interval of size nCn^{–C}

  • Ilia Binder

    University of Toronto, Toronto, Ontario, Canada
  • Mircea Voda

    The University of Chicago, USA
An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$ cover
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Abstract

We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift xx+ωx\rightarrow x+\omega. We assume that the Lyapunov exponent L(E0)L(E_{0}) of the corresponding Jacobi cocycle satisfies L(E0)γ>0L(E_{0})\ge\gamma>0. In this setting we prove that the number of eigenvalues Ej(n)(x)E_{j}^{(n)}(x) of a submatrix of size nn contained in an interval II centered at E0E_{0} with I=nC1|I|=n^{-C_{1}} does not exceed (logn)C0\left(\log n\right)^{C_{0}} for any xx. Here nn0n\ge n_{0}, and n0n_{0}, C0C_{0}, C1C_{1} are constants depending on γ\gamma (and the other parameters of the problem).

Cite this article

Ilia Binder, Mircea Voda, An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size nn contained in an interval of size nCn^{–C}. J. Spectr. Theory 3 (2013), no. 1, pp. 1–45

DOI 10.4171/JST/36