For non-Hermitian Hamiltonians, the Arnoldi diagonal and subdiagonal coefficients are the Flaschka variables of the finite two-dimensional Toda lattice, and their squares give both the Fubini-Study metric and the Berry curvature of the Krylov subspaces.
Iso-spectral deformations of general matrix and their reductions on Lie algebras
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We study an iso-spectral deformation of general matrix which is a natural generalization of the Toda lattice equation. We prove the integrability of the deformation, and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, we consider several reductions of the hierarchy. These include not only the integrable systems studied by Bogoyavlensky and Kostant, but also their generalizations which were not known to be integrable before. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.
citation-role summary
background 1
citation-polarity summary
fields
quant-ph 1years
2026 1verdicts
CONDITIONAL 1roles
background 1polarities
background 1representative citing papers
citing papers explorer
-
Two-dimensional Toda--Arnoldi correspondence: Holomorphic Krylov geometry and counterdiabatic transport
For non-Hermitian Hamiltonians, the Arnoldi diagonal and subdiagonal coefficients are the Flaschka variables of the finite two-dimensional Toda lattice, and their squares give both the Fubini-Study metric and the Berry curvature of the Krylov subspaces.