Equations in Dual Variables for Whittaker Functions

Babelon, O.

doi:10.1023/b:math.0000010714.56215.2a

Cited by 17 publications

(37 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In other words, one should find how the given operator on H Toda acts on the space H sep where the separation of variables is realised. This inverse problem has been solved for different examples of quasi-local operators and for various models [8,9,82,105,122,124,125,139].…”

Section: Multiple Integral Representationsmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

View full text Add to dashboard Cite

This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

show abstract

Section: Multiple Integral Representationsmentioning

confidence: 99%

“…The resolution of the inverse problem for the Toda chain has been pioneered by Babelon [8,9] in 2002 and further developed in the works [105,139]. These results, along with the unitarity of the separation of variables transform U lead to multiple integral representations for the form factors.…”

Section: Multiple Integral Representationsmentioning

confidence: 99%

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

View full text Add to dashboard Cite

show abstract

“…First, it could be interesting to apply quantum Hamiltonian reduction [21] to gain a better understanding of the quantum mechanical version of Toda duality. For the state of art of this subject, see the papers [22,23,17] and references therein. Second, it would be important to generalize the group-theoretic framework presented in this paper so as to accommodate the open relativistic Toda lattice, whose dual was also derived in [2].…”

Section: Group-theoretic Interpretation Of Toda Dualitymentioning

confidence: 99%

Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction

Fehér

2013

Physics Letters A

View full text Add to dashboard Cite

An alternative derivation of the known action-angle map of the standard open Toda lattice is presented based on its identification as the natural map between two gauge slices in the relevant symplectic reduction of the cotangent bundle of GL(n, R). This then permits to interpret Ruijsenaars' action-angle duality for the Toda system in the same group-theoretic framework which was established previously for Calogero type systems.

show abstract

“…In the present paper, we push forward the techniques developed by Babelon [2,3] and demonstrate that one can derive equations in dual variables for more general operators. Such operators are built out of certain subcomponents of the model's monodromy matrix.…”

Section: Introductionmentioning

confidence: 97%

Aspects of the inverse problem for the Toda chain

Kozłowski

2013

Journal of Mathematical Physics

View full text Add to dashboard Cite

We generalize Babelon's approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub-constituents of the model's monodromy matrix. Further, we also apply Sklyanin's recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators. The schemes discussed in this paper appear to be universal and thus, in principle, applicable to many models solvable through the quantum separation of variables.

show abstract

Equations in Dual Variables for Whittaker Functions

Cited by 17 publications

References 11 publications

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction

Aspects of the inverse problem for the Toda chain

Contact Info

Product

Resources

About