Boom and bust cycles due to pseudospectra of matrices with unimodular spectra

Bhat, Junaid Majeed (author), Bensa, Jaš (author), Žnidarič, Marko (author)

We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity and therefore have all eigenvalues on the unit circle. Naively, one would expect that the matrix elements of high powers of the matrix asymptotically cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices, one can have an exponential growth. For finite matrices, this exponential growth is part of a repeating cycles of exponential growth followed by an exponential decay. The effect can occur for strongly non-normal matrices if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.

• Language: English
• Keywords: pseudospectrum, non-normal matrices, exponential growth
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 15 str.
• Numbering: Vol. 58, no. 29, art. 295202
• PID: 20.500.12556/RUL-182235
• UDC: 512
• ISSN on article: 1751-8121
• DOI: 10.1088/1751-8121/aded52
• COBISS.SI-ID: 276509187
• Publication date in RUL: 05.05.2026

Record is a part of a journal

• Title: Journal of physics. A, Mathematical and theoretical
• Shortened title: J. phys., A, Math. theor.
• Publisher: IOP Publishing
• ISSN: 1751-8121
• COBISS.SI-ID: 3763226

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: psevdospekter, nenormalna matrika, eksponentna rast

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: J1-4385
• Name: Kvantna večdelčna fizika in nehermitskost

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0402
• Name: Matematična fizika