The quantum mechanical toda lattice

Gutzwiller, Martin C.

doi:10.1016/0003-4916(80)90214-6

Cited by 74 publications

(80 citation statements)

References 6 publications

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“…These occur at values of the form = 2π r s , (2.62) where r, s are coprime integers. 1 This fact was first pointed out in a closely related context in [22], based on insights from [23]. These poles are not physical, and they should be cancelled by non-perturbative contributions.…”

Section: Jhep05(2016)133mentioning

confidence: 94%

“…Alternatively, one can write these quantization conditions in terms of integral equations of the TBA type [8]. These TBA equations are obtained by resumming the instanton expansion of the partition function (see [18,19] for a detailed derivation), and as shown in [20] they are equivalent to the quantization conditions in [1][2][3][4].…”

Section: Jhep05(2016)133mentioning

confidence: 99%

“…In the case of the Toda lattice, these conditions were obtained in [1][2][3][4], by finding appropriate solutions of the one-dimensional quantum Baxter equation associated to the integrable system. The Toda lattice has a relativistic generalization [5], and the techniques of separation of variables lead to a Baxter equation involving difference operators [6,7].…”

Section: Introductionmentioning

confidence: 99%

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Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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Inspired by recent connections between spectral theory and topological string theory, we propose exact quantization conditions for the relativistic Toda lattice of N particles. These conditions involve the Nekrasov-Shatashvili free energy, which resums the perturbative WKB expansion, but they require in addition a non-perturbative contribution, which is related to the perturbative result by an S-duality transformation of the Planck constant. We test the quantization conditions against explicit calculations of the spectrum for N = 3. Our proposal can be generalized to arbitrary toric Calabi-Yau manifolds and might solve the corresponding quantum integrable system of Goncharov and Kenyon.

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Section: Jhep05(2016)133mentioning

confidence: 94%

Section: Jhep05(2016)133mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact quantization conditions for the relativistic Toda lattice

Hatsuda

Mariño

2016

J. High Energ. Phys.

172

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“…We shall refer to the general class of such integrals as the sinh model: When β = 1 and for specific choices of the constants ω 1 , ω 2 > 0 and of the confining potential W, z N represents norms or arises as a fundamental building block of certain classes of correlation functions in quantum integrable models that are solvable by the quantum separation of variable method. This method takes its roots in the works of Gutzwiller [85,86] on the quantum Toda chain and has been developed in the mid '80s by Sklyanin [137,138] as a way of circumventing certain limitations inherent to the algebraic Bethe Ansatz. Expressions for the norms or correlation functions for various models solvable by the quantum separation of variables method have been established, e.g.…”

Section: An Opening Discussionmentioning

confidence: 99%

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

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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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 ...

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“…where h k (p, q) are the Hamiltonians of the open Toda chain obtained by removing particle N + 1 from the closed chain [2].…”

Section: Introductionmentioning

confidence: 99%

Equations in Dual Variables for Whittaker Functions

Babelon

2003

Letters in Mathematical Physics

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It is known that the Whittaker functions w(q, λ) associated to the group SL(N ) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables q i . Using the expression of the q i for the closed Toda chain in terms of Sklyanin variables λ i , and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in λ i .

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The quantum mechanical toda lattice

Cited by 74 publications

References 6 publications

Exact quantization conditions for the relativistic Toda lattice

Exact quantization conditions for the relativistic Toda lattice

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

Equations in Dual Variables for Whittaker Functions

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