CMV matrices in random matrix theory and integrable systems: A survey

Journal of Physics A: Mathematical and General

Volumn 39, Issue 28, 2006, Pages 8811-8822

  Nenciu, Irina ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745537657     PISSN: 03054470     EISSN: 13616447     Source Type: Journal    
DOI: 10.1088/0305-4470/39/28/S04     Document Type: Review

References (36)

1. Ablowitz M J and Ladik J F 1975 Nonlinear differential-difference equations J. Math. Phys. 16 598-603
2. Ablowitz M J and Ladik J F 1976 Nonlinear differential-difference equations and Fourier analysis J. Math. Phys. 17 1011-8
3. Ammar G S and Gragg W B 1994 Schur flows for orthogonal Hessenberg matrices Hamiltonian and Gradient Flows, Algorithms and Control (Providence, RI: American Mathematical Society) pp 27-34 (Fields Inst. Commun. 3)
4. Babelon O, Bernard D and Talon M 2003 Introduction to classical integrable systems Cambridge Monographs on Mathematical Physics (Cambridge: Cambridge University Press)
5. Cantero M J, Moral L and Velázquez L 2003 Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle Linear Algebra Appl. 362 29-56
6. Cantero M J, Moral L and Velázquez L 2004 Minimal representations of unitary operators and orthogonal polynomials on the unit circle Preprint math.CA/0405246
7. Deift P 1996 Integrable Hamiltonian systems Dynamical Systems and Probabilistic Methods in Partial Differential Equations (Lectures in Applied Mathematics vol 31 (Providence, RI: American Mathematical Society) pp 103-38
8. Deift P, Li L-C, Nanda T and Tomei C 1986 The Toda flow on a generic orbit is integrable Commun. Pure Appl. Math. 39 183-232
9. Deift P, Li L-C and Tomei C 1985 Toda flows with infinitely many variables J. Funct. Anal. 64 358-402
10. Dumitriu I and Edelman A 2002 Matrix models for beta ensembles J. Math. Phys. 43 5830-47
11. Dyson F 1962 Statistical theory of the energy levels of complex systems: I, II, and III J. Math. Phys. 3 140-56, 157-65 and 166-75
12. Faybusovich L and Gekhtman M 1999 On Schur flows J. Phys. A: Math. Gen. 32 4671-80
13. Fermi E, Pasta J and Ulam S 1965 Studies of nonlinear problems Collected Works of Enrico Fermi vol II (Chicago: University of Chicago Press) pp 978-88
14. Flaschka H 1974 The Toda lattice: I. Existence of integrals Phys. Rev. B 9 1924-5
15. Forrester P J and Rains E M 2005 Interpretations of some parameter dependent generalizations of classical matrix ensembles Probab. Theory Relat. Fields 131 1-61
16. Forrester P J and Rains E M 2005 Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices Preprint math.PR/0505552
17. Gekhtman M 1993 Non-Abelian nonlinear lattice equations on finite interval J. Phys. A: Math. Gen. 26 6303-17
18. Gelfand I M and Dikij L A 1987 A family of Hamiltonian structures related to nonlinear integrable differential equations Izrail M. Gelfand Collected Papers vol 1 (Berlin: Springer) pp 625-46
19. Geronimus Ya L 1944 On polynomials orthogonal on the circle, on trigonometric moment-problem and on allied Carathodory and Schur functions Mat. Sb. 15 99-130 (in Russian)
20. Geronimus Ya L 1961 Orthogonal Polynomials (New York: Consultants Bureau)
21. Hénon M 1974 Integrals of the Toda lattice Phys. Rev. B 9 1921-23
22. Killip R and Nenciu I 2004 Matrix models for circular ensembles Int. Math. Res. Not. 50 2665-701
23. Killip R and Nenciu I CMV: the unitary analogue of Jacobi matrices Preprint (Commun. Pure Appl. Math. at press)
24. Kostant B 1979 The solution to a generalized Toda lattice and representation theory Adv. Math. 34 195-338
25. Li L-C 2005 Some remarks on CMV matrices and dressing orbits Preprint math.SG/0507299
26. Lippert R A 2002 A matrix model for the β-Jacobi ensemble J. Math. Phys. 44 4807-16
27. Moser J 1975 Finitely many mass points on the line under the influence of an exponential potential-an integrable system Dynamical systems, Theory and Applications (Lecture Notes in Physics vol 38) (Berlin: Springer) pp 467-97
28. Mukaihira A and Nakamura Y 2002 Schur flow for orthogonal polynomials on the unit circle and its integrable discretization J. Comput. Appl. Math. 139 75-94
29. Nenciu I 2005 Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle Int. Math. Res. Not. 11 647-86
30. Olshanetsky M A, Perelomov A M, Reyman A G and Semenov-Tian-Shansky M A Integrable systems: II. Dynamical systems: VII Encycl. Math. Sci. 16 83-259
31. Simon B 2005 Orthogonal polynomials on the unit circle: part 1. Classical theory AMS Colloquium Series (Providence, RI: American Mathematical Society)
32. Simon B 2005 Orthogonal polynomials on the unit circle: part 2. Spectral theory AMS Colloquium Series (Providence, RI: American Mathematical Society)
33. Szeg G 1975 Orthogonal polynomials American Mathematical Society Colloquium Publications vol XXIII (Providence, RI: American Mathematical Society)
34. Thiele C and Tao T The Dirac non-linear Fourier transform on (unpublished)
35. Toda M 1989 Theory of Nonlinear Lattices (Springer Series in Solid-State Sciences 20) 2nd edn (Berlin: Springer)
36. Watkins D S 1993 Some perspectives on the eigenvalue problem SIAM Rev. 35 (no. 3) 430-71