Abstract
Let (−A, B, C) be a linear system in continuous time t > 0 with input and output space \(\mathbb {C}\) and state space H. The function ϕ (x)(t) = Ce −(t+2x)A B determines a Hankel integral operator \(\Gamma _{\phi _{(x)}}\) on \(L^2((0, \infty ); \mathbb {C})\); if \(\Gamma _{\phi _{(x)}}\) is trace class, then the Fredholm determinant \(\tau (x)=\det (I+ \Gamma _{\phi _{(x)}})\) defines the tau function of (−A, B, C). Such tau functions arise in Tracy and Widom’s theory of matrix models, where they describe the fundamental probability distributions of random matrix theory. Dyson considered such tau functions in the inverse spectral problem for Schrödinger’s equation − f″ + uf = λf, and derived the formula for the potential \(u(x)=-2{{d^2}\over {dx^2}}\log \tau (x)\) in the self-adjoint scattering case (Commun Math Phys 47:171–183, 1976). This paper introduces a operator function R x that satisfies Lyapunov’s equation \({{dR_x}\over {dx}}=-AR_x-R_xA\) and \(\tau (x)=\det (I+R_x)\), without assumptions of self-adjointness. When − A is sectorial, and B, C are Hilbert–Schmidt, there exists a non-commutative differential ring \({\mathcal A}\) of operators in H and a differential ring homomorphism \(\lfloor \,\,\rfloor :{\mathcal A}\to \mathbb {C}[u,u', \dots ]\) such that u = −4⌊A⌋, which extends the multiplication rules for Hankel operators considered by Pöppe and McKean (Cent Eur J Math 9:205–243, 2011).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. Aktosun, F. Demontis, C. van der Mee, Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Prob. 23, 2171–2195 (2007)
M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra (Addison Wesley, Reading, 1969)
H.F. Baker, Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions (Cambridge University Press, Cambridge, 1995)
R. Bhatia, P. Rosenthal, How and why to solve the operator equation AX − XB = Y . Bull. Lond. Math. Soc. 29, 1–21 (1997)
G. Blower, Linear systems and determinantal random point fields. J. Math. Anal. Appl. 335, 311–334 (2009)
G. Blower, On tau functions for orthogonal polynomials and matrix models. J. Phys. A 44, 285202 (2011)
Y.V. Brezhnev, What does integrability of finite-gap or soliton potentials mean? Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366, 923–945 (2008)
F.J. Dyson, Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976)
K.-J. Engel, R. Nagel, One Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000)
N. Ercolani, H.P. McKean, Geometry of KdV. IV: Abel sums, Jacobi variety and theta function in the scattering case. Invent. Math. 99, 483–544 (1990)
I.M. Gelfand, L.A. Dikii, Integrable nonlinear equations and the Liouville theorem. Funct. Anal. Appl. 13, 6–15 (1979)
I.M. Gelfand, B.M. Levitan, On the determination of a differential equation from its spectral function. Izvestiya Akad. Nauk SSSR Ser. Mat. 15, 309–360 (1951)
F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions Volume I: (1 + 1)-Dimensional Continuous Models (Cambridge University Press, Cambridge, 2003)
F. Gesztesy, B. Simon, The Xi function. Acta Math. 176, 49–71 (1996)
J.A. Goldstein, Semigroups of Linear Operators and Applications (Oxford University Press, Oxford, 1985)
S. Grudsky, A. Rybkin, On classical solutions of the KdV equation. Proc. Lond. Math. Soc. 121, 354–371 (2020)
S. Grudsky, A. Rybkin, Soliton theory and Hankel operators. SIAM J. Math. Anal. 47, 2283–2323 (2015)
E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, 1968)
V.G. Kac, Infinite Dimensional Lie Algebras (Cambridge University Press, Cambridge, 1985)
V. Katsnelson, D. Volok, Rational solutions of the Schlesinger system and isoprincipal deformations of rational matrix functions. I. Oper. Theory Adv. Appl. 149, 291–348 (2004)
S. Kotani, Construction of KdV flow I. τ-function via Weyl function. Zh. Mat. Fiz. Anal. Geom. 14, 297–335 (2018)
I.M. Krichever, The integration of nonlinear equations by the methods of algebraic geometry. Funct. Anal. Appl. 11, 12–26 (1977)
V.B. Matveev, Darboux transformation and explicit solutions of the Kadomtcev–Petviaschvily equation, depending upon functional parameters. Lett. Math. Phys. 3, 213–216 (1979)
H.P. McKean, Fredholm determinants. Cent. Eur. J. Math. 9, 205–243 (2011)
A.V. Megretskii, V.V. Peller, S.R. Treil, The inverse spectral problem for self-adjoint Hankel operators. Acta Math. 174, 241–309 (1995)
T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries, and Infinite Dimensional Algebras (Cambridge University Press, Cambridge, 2000)
M. Mulase, Cohomological structure in soliton equations and Jacobian varieties. J. Differ. Geom. 19, 403–430 (1984)
N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 1 (American Mathematical Society, Providence, 2002)
S. Novikov, S.V. Manakov, L.P. Pitaevskii, V.F. Zakharov, Theory of Solitons, the Inverse Scattering Method (Consultants Bureau, New York and London, 1984)
V.V. Peller, Hankel Operators and Their Applications (Springer, New York, 2003)
C. Pöppe, The Fredholm determinant method for the KdV equations. Phys. D 13, 137–160 (1984)
C. Pöppe, D.H. Sattinger, Fredholm determinants and the τ function for the Kadomtsev–Petviashvili hierarchy. Publ. Res. Inst. Math. Sci. 24, 505–538 (1988)
M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations (Springer, Berlin, 2003)
A. Rybkin, The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line. Nonlinearity 24, 2953–2990 (2011)
G. Segal, G. Wilson, Loop groups and equations of KdV type. Inst. Hautes Études Sci. Publ. Math. 61, 5–65 (1985)
I.N. Sneddon, The Use of Integral Transforms (McGraw-Hill, New York, 1972)
C.A. Tracy, H. Widom, Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994)
M. Trubowitz, The inverse problem for periodic potentials. Commun. Pure Appl. Math. 30, 321–337 (1977)
T. Zhang, S. Venakides, Periodic limit of inverse scattering. Commun. Pure Appl. Math. 46, 819–865 (1993)
Acknowledgements
GB thanks Henry McKean for helpful conversations. SLN thanks EPSRC for financially supporting this research. The authors thank the referee for drawing attention to recent literature.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Blower, G., Newsham, S.L. (2021). Tau Functions Associated with Linear Systems. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-51945-2_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-51944-5
Online ISBN: 978-3-030-51945-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)