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According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes:

  • periodicity ~ potential well or lattice (order)

  • lack of absolutely continued spectrum ~ Anderson localization(disorder)

Given this, what is the corresponding spectrum of a particle in quasi-order materials (like quasicrystals，quasi-order)? or what is about corresponding power series?

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Quasiperiodic potentials, such as the Fibonacci chain, have a spectrum that is called singular continuous, with a fractal structure, see

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    $\begingroup$ Your opening sentence is incorrect, other types of spectrum (such as absolutey continuous or pure point spectrum) are also possible for quasi-periodic potentials. $\endgroup$
    – Christian Remling
    Commented Jul 6, 2017 at 19:06
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    $\begingroup$ Also, the potential of the Fibonacci Hamiltonian is not quasi-periodic. I think you confused quasicrystals and quasi-periodic sequences. $\endgroup$
    – Christian Remling
    Commented Jul 6, 2017 at 19:13
  • $\begingroup$ @ChristianRemling why do you not clarify these ideas and answer the questions? $\endgroup$
    – XL _At_Here_There
    Commented Jul 6, 2017 at 21:20
  • $\begingroup$ @XL_at_China: To be totally honest, I find your question hard to answer (and understand, in fact) since it's so very general and vague. If you just wanted more material, then Carlo's answer provides that. Sorry for the unsolicited advice, but in my experience at least, concrete, to the point questions tend to work better on this site. $\endgroup$
    – Christian Remling
    Commented Jul 7, 2017 at 0:28
  • $\begingroup$ @ChristianRemling Surely I want more material. But one question is about your article and Simon's article. One conclusion is related to power series: if the coefficients $a_i$are finite($a_i \in A, A is finite$), the sequence of the coefficients $[a_0,a_1,a_2.....]$ will be eventually periodic or random. I think possibly what randomness means in your article, is it consistent with Kolmogorov's definition or Chaitin's definition? Maybe it includes more than strict random sequence in the Kolmogorov's randomness sense. About the randomness, see Kolmogorov, Chaitin, or [to be continued] $\endgroup$
    – XL _At_Here_There
    Commented Jul 7, 2017 at 1:07

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