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  Singular spectrum in physics

+ 4 like - 0 dislike

In general, the spectrum of self-adjoint operators in a Hilbert space decomposes into a discrete, an absolutely continuous, and a singular spectrum. In nonrelativistic multiparticle scattering, the absence of the singular spectrum can be proved in the physically relevant situation. 

Are there areas of physics where the singular spectrum is nontrivial? If yes, what is its physical interpretation? 

asked Aug 28, 2016 in Theoretical Physics by Arnold Neumaier (15,787 points) [ revision history ]
edited Aug 28, 2016 by Arnold Neumaier

1 Answer

+ 2 like - 0 dislike

I would assume that the work of Barry Simon (known e.g. for Reed/Simon, Methods of Modern Mathematical Physics) is interesting for you.

Regarding his  "Singular Continuous Spectrum Revolution" see e.g. https://www.ma.utexas.edu/mp_arc/c/06/06-181.pdf.

A physically relevant problem is the almost Mathieu operator (cf. the pdf cited, page 11) which can be used (for appropriate parameters) to model the Quantum Hall effect and one-dimensional quasiperiodic chains.  

Physical results can be anomalous diffusion properties (exponents connected to generalizes dimensions of the spectrum), see e.g. https://arxiv.org/pdf/cond-mat/9811323.pdf.

Here is another paper showing that (in a certain, well defined sense) singular continuous spectra are generic: https://arxiv.org/abs/math/9410217.

answered Dec 3, 2016 by Andreas Rüdinger (90 points) [ revision history ]
edited Dec 3, 2016 by Andreas Rüdinger

Excellent! Before seeing your answer I indeed had the impression that (as in the treatise by Reed & Simon) the singual spectrum was an anomaly - just the setting that Yoram Last (your first link) contrasts with the interesting developments since this treatise was written. 

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