Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent

Jimbo, Michio; Miwa, Tetsuji; Môri, Yasuko; Sato, Masahide

doi:10.1016/0167-2789(80)90006-8

Cited by 462 publications

(607 citation statements)

References 34 publications

Supporting

Mentioning

582

Contrasting

Unclassified

Order By: Relevance

“…By the one-to-one correspondence between the solutions of (2.4) and the monodromy data [26], we therefore have Theorem 5 Let Ψ obey the reality condition (3.38) and let the monodromy matrices satisfy the relations (3.49). Then the constants of integration in (3.6) and (2.7) may be chosen in such a way that g(ξ,ξ) solves the Ernst equation (2.4), is real and symmetric, and the conformal factor h is real.…”

Section: Theoremmentioning

confidence: 99%

“…Our equations (3.22) are closely related to the so-called Schlesinger equations [45] which play an important role in the theory of integrable systems [26]. To exhibit the relation, let us consider γ j , j = 1, ..., N as independent deformation parameters and suppose that the monodromy data {T j , C j } are γ j -independent.…”

Section: Relation To the Schlesinger Equationsmentioning

confidence: 99%

“…The closure condition dω 0 = 0 may be directly verified by use of (3.22) and (3.2). Following the general prescription given in [26], we have…”

Section: Deformation Equations and τ -Functionmentioning

confidence: 99%

“…Following [26] we will refer to the set {w j , T j , C j , L j , M j (w)} as the set of monodromy data of the function Ψ(γ) and the associated solution g(ξ,ξ) of (2.4). The logarithmic derivative ("spectral parameter current")…”

mentioning

confidence: 99%

See 3 more Smart Citations

Isomonodromic quantization of dimensionally reduced gravity

Korotkin¹,

Nicolai²

1996

Nuclear Physics B

View full text Add to dashboard Cite

Abstract. We present a detailed account of the isomonodromic quantization of dimensionally reduced Einstein gravity with two commuting Killing vectors. This theory constitutes an integrable "midi-superspace" version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation of variables in the isomonodromic sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are thereby reduced to the Knizhnik-Zamolodchikov equations for SL(2, R). The physical states are shown to live in a well defined Hilbert space and are manifestly invariant under the full diffeomorphism group. An infinite set of independent observablesà la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of SL(2, R), we construct explicit quantum states. However, satisfying the additional constraints associated with the coset space SL(2, R)/SO(2) requires solutions based on the principal series representations, which are not yet known. We briefly discuss the possible implications of our results for string theory.

show abstract

Section: Theoremmentioning

confidence: 99%

Section: Relation To the Schlesinger Equationsmentioning

confidence: 99%

“…The closure condition dω 0 = 0 may be directly verified by use of (3.22) and (3.2). Following the general prescription given in [26], we have…”

Section: Deformation Equations and τ -Functionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Isomonodromic quantization of dimensionally reduced gravity

Korotkin¹,

Nicolai²

1996

Nuclear Physics B

View full text Add to dashboard Cite

show abstract

“…where 5) and Z is a certain constant whose value will be defined shortly. Here φ x is inverse Fourier transform of φ(ξ), and φ ± (ξ) are the Wiener-Hopf factors satisfying; 6) and also satisfying the condition that the functions φ ± when extended away from R are analytic in the upper and lower half-plane respectively.…”

Section: Wiener-hopf Determinantsmentioning

confidence: 99%

A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity

Basor¹,

Chen²

2003

Integral Equations and Operator Theory

View full text Add to dashboard Cite

The continuous analogue of a Toeplitz determinant identity for Wiener-Hopf operators is proved. An example which arises from random matrix theory is studied and an error term for the asymptotics of the determinant is computed. Wiener-Hopf DeterminantsRecently, a beautiful identity due to Borodin and Okounkov was proved for Toeplitz determinants which shows how one can write a Toeplitz determinant as a Fredholm determinant. In this note we generalize this to the Wiener-Hopf case. The proof in the Wiener-Hopf case follows identically with the second one given in [1]. We include it here for completeness sake and because the nature of the identity is slightly different in the continuous verses discrete convolution setting.In the Wiener-Hopf case we begin with a Fredholm determinant on a finite interval and then show how this can be written as a Fredholm determinant of an operator defined on L 2 of a half-line. The point is the second operator has a very "small" kernel and thus higher order approximations (as a function of the length of the finite interval) can be found.We now state the analogue of the identity and then apply it to a particular case to show how error estimates can be computed. In the future we hope to refine the estimates given here, apply this identity to other important examples, and also extend it to other operators.We consider the Fredholm determinant of the finite Wiener-Hopf operator det(I − K [0,α] ), (1.1)

show abstract