Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler–Cassini identity.

Download Full-text

Generalized Gibbs Ensembles of the Classical Toda Chain

Journal of Statistical Physics ◽

10.1007/s10955-019-02320-5 ◽

2019 ◽

Vol 180 (1-6) ◽

pp. 4-22 ◽

Cited By ~ 14

Author(s):

Herbert Spohn

Keyword(s):

Toda Chain

Download Full-text

Exact holographic RG flows and the A1 × A1 Toda chain

Journal of High Energy Physics ◽

10.1007/jhep05(2019)117 ◽

2019 ◽

Vol 2019 (5) ◽

Cited By ~ 3

Author(s):

Irina Ya. Aref’eva ◽

Anastasia A. Golubtsova ◽

Giuseppe Policastro

Keyword(s):

Toda Chain

Download Full-text

Lecture on SUSY Gauge Theories and Integrable Systems

International Journal of Modern Physics B ◽

10.1142/s0217979297001519 ◽

1997 ◽

Vol 11 (26n27) ◽

pp. 3093-3124

Author(s):

A. Marshakov

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Integrable Systems ◽

Gauge Theories ◽

Toda Chain ◽

Quantum Field ◽

Standard Quantum ◽

Integrable Equations ◽

Complex Curves ◽

Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

Download Full-text

The Toda chain as a reduced system

Theoretical and Mathematical Physics ◽

10.1007/bf01047139 ◽

1980 ◽

Vol 45 (1) ◽

pp. 843-854 ◽

Cited By ~ 12

Author(s):

M. A. Ol'shanetskii ◽

A. M. Perelomov

Keyword(s):

Toda Chain

Download Full-text

Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202105059 ◽

2002 ◽

Vol 31 (9) ◽

pp. 513-553 ◽

Cited By ~ 3

Author(s):

Stanislav Pakuliak ◽

Sergei Sergeev

Keyword(s):

Bethe Ansatz ◽

Weyl Algebra ◽

Separation Of Variables ◽

Toda Chain ◽

Special Choice ◽

Finite Dimensional Representation ◽

Classical Counterpart ◽

Discrete Integrable System ◽

Root Of Unity ◽

Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

Download Full-text