BERKOLAIKO, G. and WINN, B., 2018. Maximal scarring for eigenfunctions of quantum graphs. Nonlinearity, 31 (10), 4812.
We prove the existence of scarred eigenstates for star graphs with scattering matrices
at the central vertex which are either a Fourier transform matrix, or a matrix that
prohibits back-scattering. We prove the existence of scars that are half-delocalised on
a single bond. Moreover we show that the scarred states we construct are maximal
in the sense that it is impossible to have quantum eigenfunctions with a significantly
lower entropy than our examples.
These scarred eigenstates are on graphs that exhibit generic spectral statistics of
random matrix type in the large graph limit, and, in contrast to other constructions,
correspond to non-degenerate eigenvalues; they exist for almost all choices of lengths
This paper is in closed access until 12th September 2019.
GB acknowledges partial support from the NSF under grant DMS1410657.