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.