CMV: The unitary analogue of Jacobi matrices

Rowan Killip, Irina Nenciu

Type: Article

Publication Date: 2006-08-28

Citations: 80

DOI: https://doi.org/10.1002/cpa.20160

Abstract

Abstract We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velázquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well‐known properties of Jacobi matrices: foliation by co‐adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz‐Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz‐Ladik hierarchy and describe the long‐time behavior of this system. © 2006 Wiley Periodicals, Inc.

Locations

Communications on Pure and Applied Mathematics - View
arXiv (Cornell University) - View - PDF

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Works Cited by This (22)

Action	Title	Year	Authors
+	Symplectic Aspects of Some Eigenvalue Algorithms	1993	Percy Deift L.‐C. Li Carlos Tomei
+	Integrable Systems of Classical Mechanics and Lie Algebras	1990	A. M. Perelomov
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+	Some Perspectives on the Eigenvalue Problem	1993	David S. Watkins
+ PDF Chat	Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices	2006	Peter J. Forrester Eric M. Rains
+	The solution to a generalized Toda lattice and representation theory	1979	Bertram Kostant
+	Toda flows with infinitely many variables	1985	Percy Deift L.-C. Li Carlos Tomei
+	On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-devries type equations	1978	Mark Adler