Related papers: Toda Systems, Cluster Characters, and Spectral Net…
Characteristic integrals of Toda field theories associated to simple Lie algebras are presented in the most explicit forms, both in terms of the formulas and in terms of the proofs.
This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations,…
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies…
The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear…
Hamiltonians of a wide-spread class of strongly coupled quantum system models are expressed as nonlinear functions of $sl(2)$ generators. It enables us to use the $sl(2)$ formalism, in particular, $sl(2)$ generalized coherent states (GCS)…
The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the N-th powers of the quantum operators of the corresponding quantum integrable models.…
We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a…
We consider different phase spaces for the Toda flows and the less familiar SVD flows. For the Toda flow, we handle symmetric and non-symmetric matrices with real simple eigenvalues, possibly with a given profile. Profiles encode, for…
Following a prescription of \cite{4} for a solitonic specialization of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely…
A non-standard generalisation of the Bender potentials $x^2(\ii x^\ve)$ is suggested. The spectra are obtained numerically and some of their particular properties are discussed.
The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one…
The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$…
I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of…
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…
We endow Ruijsenaars' open difference Toda chain with a one-sided boundary interaction of Askey-Wilson type and diagonalize the quantum Hamiltonian by means of deformed hyperoctahedral $q$-Whittaker functions that arise as a $t=0$…
High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics…
We systematically study inhomogeneous Hamiltonians in two-dimensional conformal field theories within the framework of the AdS/CFT correspondence by relating them to two-dimensional curved backgrounds. We propose a classification of…
For each one of the Lie algebras $\mathfrak{gl}_{n}$ and $\widetilde {\mathfrak{gl}}_{n}$, we constructed a family of integrable generalizations of the Toda chains characterized by two integers $m_{+}$ and $m_{-}$. The Lax matrices and the…
The periodic Toda lattice with $N$ sites is globally symplectomorphic to a two parameter family of $N-1$ coupled harmonic oscillators. The action variables fill out the whole positive quadrant of $\R^{N-1}$. We prove that in the interior of…
Ordinary and gl(n,R) generalized Toda systems as well as a related hierarchy are probed with respect to certain quantization characteristics. "Quantum" canonical and Poisson transformations are used to study quantizations of transformed…