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Spectral properties and the dynamics

We review in this section some spectral properties which can be deduced from Fourier coefficients tex2html_wrap_inline746 of spectral measures.

The discrete spectrum. Wiener's theorem in Fourier theory

displaymath748

allows to determine, whether there is some discrete part in the spectral measure tex2html_wrap_inline534 and so some eigenvalues of L. This tool for detecting point spectrum is used in quantum dynamics (see [1]). If the potential takes finitely many (rational) values, then tex2html_wrap_inline754 is a (rational) number which can be computed exactly. Evolution (2.1) allows so to treat the evolution of any bounded discrete Schrödinger operators in one dimension with the same efficiency as kicked quantum oscillators or kicked Harper models.

The tex2html_wrap_inline756 -absolutely continuous spectrum. If there exists a constant C, such that tex2html_wrap_inline760 then tex2html_wrap_inline762 , because of Plancherel's theorem tex2html_wrap_inline764 with tex2html_wrap_inline766 . It is more difficult to detect other absolutely continuous spectrum. While tex2html_wrap_inline768 goes to zero by the Riemann-Lebesgue lemma, if tex2html_wrap_inline770 is absolutely continuous, the decay can be arbitary slow and decay is also possible if tex2html_wrap_inline770 is singular continuous. The tex2html_wrap_inline756 -absolutely continuous spectrum is important because by a result of Kato, the closure of all vectors tex2html_wrap_inline678 with tex2html_wrap_inline756 -absolutely continuous spectral measures tex2html_wrap_inline534 is the absolutely continuous subspace.

Singular continuous spectrum. If tex2html_wrap_inline782 does not converge to zero, then by the Riemann-Lebesgue theorem, tex2html_wrap_inline534 must have some singular spectrum. If also tex2html_wrap_inline786 converges to zero, then L has purely singular continuous spectrum, a property, which is generic in many situations (see [25, 27]). However, if L has purely singular continuous spectrum, it is still possible that tex2html_wrap_inline792 . The question, whether tex2html_wrap_inline782 converges to zero or not can be subtle and there are both singular continuous measures for which tex2html_wrap_inline782 does or does not converge to zero. Singular continuous spectrum occurs often in solid state physics. The dynamics of U on the singular continuous subspace is the least understood. It follows from Wiener's theorem and Birkhoff's ergodic theorem that the topological entropy of an unitary operator acting on the weakly compact unit ball is zero [16].

Weak continuity properties. A discrete version of a result of Stricharz [28] tells that if there exists a constant C and a function tex2html_wrap_inline802 with h(0)=0 such that tex2html_wrap_inline806 for all intervals [a,b] with length ;SPMlt;1 on the circle tex2html_wrap_inline812 (here identified with tex2html_wrap_inline814 ), then

  equation184

for all n, where tex2html_wrap_inline818 is a constant independent of anything. By a converse of Last [19], if Equation (4.1) is satisfied, then tex2html_wrap_inline820 for all intervals [a,b]. In this sense, Hölder continuity properties of the distribution function tex2html_wrap_inline824 can be detected by computing tex2html_wrap_inline782 . For recent developments in the quantum dynamics of operators with singular continuous spectrum see [9, 5, 10, 19].

Hausdorff dimension. The tex2html_wrap_inline606 -energy of a spectral measure tex2html_wrap_inline770 on tex2html_wrap_inline680 is tex2html_wrap_inline834 with tex2html_wrap_inline836 . A measure tex2html_wrap_inline838 has finite tex2html_wrap_inline606 -energy, if and only if tex2html_wrap_inline842 ([13]). The Hausdorff dimension of tex2html_wrap_inline770 , is the minimum of all Hausdorff dimensions of Borel sets S satisfying tex2html_wrap_inline848 . It is bigger or equal to tex2html_wrap_inline606 if tex2html_wrap_inline770 has finite tex2html_wrap_inline606 -energy (see Theorem 4.13 in [6]). By finding out, where the energy blows up, a lower bound on the Hausdorff dimension of tex2html_wrap_inline770 can be established and so a lower bound on the Hausdorff dimension of the support of tex2html_wrap_inline770 can be obtained.


next up previous
Next: Quantum dynamics versus random Up: A remark on quantum Previous: The Fourier coefficients of

Oliver Knill
Tue Aug 18, 1998