Sets of Non-Lyapunov Behaviour for Scalar and Matrix Schrödinger Cocycles

Ilya Goldsheid, Sasha Sodin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence pn there is a residual set of energies in the spectrum on which the middle singular value (the W-th out of 2W) grows no faster than pn1. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function ρ(t) such that ρ(t)/t is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each k, the average of the first k Lyapunov exponents is the logarithmic potential of a probability measure.

Original languageEnglish
Pages (from-to)7421-7444
Number of pages24
JournalInternational Mathematics Research Notices
Issue number9
StatePublished - 1 May 2024
Externally publishedYes


FundersFunder number
Morris Belkin Visiting Professorship at the Weizmann Institute of Science
Royal SocietyWM170012
Royal Society
Leverhulme TrustPLP-2020-064
Leverhulme Trust


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