The Toda lattice, old and new

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December 2013, 5(4): 511-530. doi: 10.3934/jgm.2013.5.511

The Toda lattice, old and new

Carlos Tomei ^1,

1.	Departamento de Matemática, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro 22451-900, Brazil

Received June 2013 Revised October 2013 Published December 2013

Abstract
Related Papers

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra.
The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.

Keywords: QR algorithm, isospectral manifolds, Toda lattice, inverse scattering., completely integrable systems.

Mathematics Subject Classification: Primary: 65F15, 37S35; Secondary: 53D0.

Citation: Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511

References:

[1]	M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219. doi: 10.1007/BF01410079.
[2]	M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[3]	M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249-315.
[4]	R. Abraham and J. Marsden, Foundations of Mechanics, Second edition, Addison-Wesley, Redwood City, CA, 1987.
[5]	R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Math. Surveys and Monographs, 28, AMS, Providence, RI, 1988.
[6]	A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra, Duke Math. J., 61 (1990), 41-65. doi: 10.1215/S0012-7094-90-06103-4.
[7]	P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics, SIAM J. Num. Anal., 28 (1991), 1463-1516. doi: 10.1137/0728076.
[8]	P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math., 39 (1986), 183-232. doi: 10.1002/cpa.3160390203.
[9]	P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math., 42 (1989), 443-521. doi: 10.1002/cpa.3160420405.
[10]	P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions, Memoirs of the Amer. Math. Soc., 100 (1992). doi: 10.1090/memo/0479.
[11]	P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables, J. Funct. Anal., 64 (1985), 358-402. doi: 10.1016/0022-1236(85)90065-5.
[12]	P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem, SIAM J. Num. Anal., 20 (1983), 1-22. doi: 10.1137/0720001.
[13]	P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463-468. doi: 10.1137/0612033.
[14]	J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971446.
[15]	J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices, SIAM J. Stat. Comput., 11 (1990), 873-912. doi: 10.1137/0911052.
[16]	H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B (3), 9 (1974), 1924-1925. doi: 10.1103/PhysRevB.9.1924.
[17]	D. Fried, The cohomology of an isospectral flow, Proc. Amer. Math. Soc., 98 (1986), 363-368. doi: 10.1090/S0002-9939-1986-0854048-6.
[18]	L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction, Jour. Geom. Phys., 21 (1997), 97-135. doi: 10.1016/S0393-0440(96)00010-1.
[19]	C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Letter., 19 (1967), 1095-1097.
[20]	V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.
[21]	T. Kapeller and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003.
[22]	B. Kostant, Quantization and representation theory, in Representation Theory of Lie Groups (ed. M. Atiyah), SRC/LMS Res. Symp. Oxford 1977, LMS Lecture Notes Series, 34, Cambridge, 1979, 287-316.
[23]	Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint, arXiv:0805.1389v1, 2008.
[24]	I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv., 32 (1977), 185-213.
[25]	I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations, Russ. Math. Surv., 35 (1980), 53-79.
[26]	P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.
[27]	R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type, Lin. Alg. Appl., 286 (1999), 149-173. doi: 10.1016/S0024-3795(98)10169-6.
[28]	R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds, Lin. Alg. Appl., 429 (2008), 387-402. doi: 10.1016/j.laa.2008.03.001.
[29]	R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence, Found. Comp. Math., 10 (2010), 15-36. doi: 10.1007/s10208-009-9047-3.
[30]	R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies, Int. Math. Res. Notices, 2013 (2013), 4382-4412. doi: 10.1093/imrn/rns186.
[31]	R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation, Lin. Alg. Appl., 361 (2003), 223-243. doi: 10.1016/S0024-3795(02)00463-9.
[32]	J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Physics, 38, Springer-Verlag, Berlin, 1975, 467-497.
[33]	J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.
[34]	B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1980.
[35]	A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.
[36]	J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure and Applied Mathematics, 130, Academic Press, Boston, MA, 1987.
[37]	S. N. M. Ruijsenaars, Relativistic Toda systems, Comm. Math. Phys., 133 (1990), 217-247. doi: 10.1007/BF02097366.
[38]	A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), Encyclopedia of Mathematical Sciences, Vol. 16, Springer-Verlag, New York, 1994.
[39]	W. Symes, Hamiltonian group actions and integrable systems, Physica D, 1 (1980), 339-374. doi: 10.1016/0167-2789(80)90017-2.
[40]	W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275. doi: 10.1016/0167-2789(82)90069-0.
[41]	N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., ().
[42]	M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan, 23 (1967), 501-506.
[43]	C. Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J., 51 (1984), 981-996. doi: 10.1215/S0012-7094-84-05144-5.
[44]	L. N.Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9780898719574.
[45]	P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81. doi: 10.1007/BF01418827.
[46]	D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows, SIAM J. Matrix Anal. Appl., 11 (1990), 301-311. doi: 10.1137/0611020.

show all references

References:

[1]	M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219. doi: 10.1007/BF01410079.
[2]	M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14 (1982), 1-15. doi: 10.1112/blms/14.1.1.
[3]	M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249-315.
[4]	R. Abraham and J. Marsden, Foundations of Mechanics, Second edition, Addison-Wesley, Redwood City, CA, 1987.
[5]	R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Math. Surveys and Monographs, 28, AMS, Providence, RI, 1988.
[6]	A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra, Duke Math. J., 61 (1990), 41-65. doi: 10.1215/S0012-7094-90-06103-4.
[7]	P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics, SIAM J. Num. Anal., 28 (1991), 1463-1516. doi: 10.1137/0728076.
[8]	P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math., 39 (1986), 183-232. doi: 10.1002/cpa.3160390203.
[9]	P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math., 42 (1989), 443-521. doi: 10.1002/cpa.3160420405.
[10]	P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions, Memoirs of the Amer. Math. Soc., 100 (1992). doi: 10.1090/memo/0479.
[11]	P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables, J. Funct. Anal., 64 (1985), 358-402. doi: 10.1016/0022-1236(85)90065-5.
[12]	P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem, SIAM J. Num. Anal., 20 (1983), 1-22. doi: 10.1137/0720001.
[13]	P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463-468. doi: 10.1137/0612033.
[14]	J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971446.
[15]	J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices, SIAM J. Stat. Comput., 11 (1990), 873-912. doi: 10.1137/0911052.
[16]	H. Flaschka, The Toda lattice. I. Existence of integrals, Phys. Rev. B (3), 9 (1974), 1924-1925. doi: 10.1103/PhysRevB.9.1924.
[17]	D. Fried, The cohomology of an isospectral flow, Proc. Amer. Math. Soc., 98 (1986), 363-368. doi: 10.1090/S0002-9939-1986-0854048-6.
[18]	L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction, Jour. Geom. Phys., 21 (1997), 97-135. doi: 10.1016/S0393-0440(96)00010-1.
[19]	C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Letter., 19 (1967), 1095-1097.
[20]	V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.
[21]	T. Kapeller and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas, 3rd Series, A Series of Modern Surveys in Mathematics], 45, Springer-Verlag, Berlin, 2003.
[22]	B. Kostant, Quantization and representation theory, in Representation Theory of Lie Groups (ed. M. Atiyah), SRC/LMS Res. Symp. Oxford 1977, LMS Lecture Notes Series, 34, Cambridge, 1979, 287-316.
[23]	Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint, arXiv:0805.1389v1, 2008.
[24]	I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv., 32 (1977), 185-213.
[25]	I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations, Russ. Math. Surv., 35 (1980), 53-79.
[26]	P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.
[27]	R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type, Lin. Alg. Appl., 286 (1999), 149-173. doi: 10.1016/S0024-3795(98)10169-6.
[28]	R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds, Lin. Alg. Appl., 429 (2008), 387-402. doi: 10.1016/j.laa.2008.03.001.
[29]	R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence, Found. Comp. Math., 10 (2010), 15-36. doi: 10.1007/s10208-009-9047-3.
[30]	R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies, Int. Math. Res. Notices, 2013 (2013), 4382-4412. doi: 10.1093/imrn/rns186.
[31]	R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation, Lin. Alg. Appl., 361 (2003), 223-243. doi: 10.1016/S0024-3795(02)00463-9.
[32]	J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Physics, 38, Springer-Verlag, Berlin, 1975, 467-497.
[33]	J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.
[34]	B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1980.
[35]	A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.
[36]	J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure and Applied Mathematics, 130, Academic Press, Boston, MA, 1987.
[37]	S. N. M. Ruijsenaars, Relativistic Toda systems, Comm. Math. Phys., 133 (1990), 217-247. doi: 10.1007/BF02097366.
[38]	A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), Encyclopedia of Mathematical Sciences, Vol. 16, Springer-Verlag, New York, 1994.
[39]	W. Symes, Hamiltonian group actions and integrable systems, Physica D, 1 (1980), 339-374. doi: 10.1016/0167-2789(80)90017-2.
[40]	W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275. doi: 10.1016/0167-2789(82)90069-0.
[41]	N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., ().
[42]	M. Toda, Wave propagation in anharmonic lattices, J. Phys. Soc. Japan, 23 (1967), 501-506.
[43]	C. Tomei, The topology of isospectral manifolds of tridiagonal matrices, Duke Math. J., 51 (1984), 981-996. doi: 10.1215/S0012-7094-84-05144-5.
[44]	L. N.Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9780898719574.
[45]	P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81. doi: 10.1007/BF01418827.
[46]	D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows, SIAM J. Matrix Anal. Appl., 11 (1990), 301-311. doi: 10.1137/0611020.

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