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# Spectral properties and the dynamics

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

The discrete spectrum. Wiener's theorem in Fourier theory

allows to determine, whether there is some discrete part in the spectral measure 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 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 -absolutely continuous spectrum. If there exists a constant C, such that then , because of Plancherel's theorem with . It is more difficult to detect other absolutely continuous spectrum. While goes to zero by the Riemann-Lebesgue lemma, if is absolutely continuous, the decay can be arbitary slow and decay is also possible if is singular continuous. The -absolutely continuous spectrum is important because by a result of Kato, the closure of all vectors with -absolutely continuous spectral measures is the absolutely continuous subspace.

Singular continuous spectrum. If does not converge to zero, then by the Riemann-Lebesgue theorem, must have some singular spectrum. If also 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 . The question, whether converges to zero or not can be subtle and there are both singular continuous measures for which 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 with h(0)=0 such that for all intervals [a,b] with length ;SPMlt;1 on the circle (here identified with ), then

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

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

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

Oliver Knill
Tue Aug 18, 1998