Hamiltonian dynamics, classical R-matrices and isomonodromic deformations

J. Harnad, Gert Sabidussi, Pavel Winternitz A

On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, $R$-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's $Q$-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.

Mark Linne

Razvan Teodorescu

The diagonal case of the $sl(2)$ Richardson-Gaudin quantum pairing model \cite{Richardson1,Richardson2,Richardson3,Richardson4,Richardson5,Richardson6,G audin76} is known to be solvable as an Abel-Jacobi inversion problem \cite{SOV,Kuznetzov,Kuz1,Kuz2,Kuz3,Kuz4,Kuz5,YAKE04}. This is an isospectral (stationary) solution to a more general integrable hierarchy, in which the full time evolution can be written as isomonodromic deformations. Physically, the more general solution is appropriate when the single-particle electronic spectrum is subject to external perturbations. The asymptotic behavior of the nonlinear oscillations in the case of elliptic solutions is derived.

Thomas Dittrich

Kazutaka Takahashi, Pratik Nandy, Adolfo del Campo

The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method for Hamiltonian deformations, which provides a systematic way of constructing solvable models from known instances. In doing so, we relate the evolution of deformed and undeformed theories and investigate their complexity. For a certain class of deformations, the resulting Krylov subspace is unchanged, and we observe time evolutions with a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones. As possible applications, we discuss coherent Gibbs states for thermodynamic systems, for which we analyze the survival probability, spread complexity, Krylov entropy, and associated time-averaged quantities. We further discuss the statistical properties of random matrices and supersymmetric systems for quadratic deformations.

Maciej Błaszak