An integral representation for the product of Airy functions

We derive a Moyal dynamical equation that describes exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a hydrodynamic theory. The question at hand is whether some large-time corrections are captured by higher-order hydrodynamics. We consider in particular the dynamics after that two chains, prepared in different conditions, are joined together. In these situations a light cone, separating regions with macroscopically different properties, emerges from the junction. In free fermionic systems some observables close to the light cone follow a universal behavior, known as Tracy-Widom scaling. Universality means weak dependence on the system's details, so this is the perfect setting where hydrodynamics could emerge. For the transverse-field Ising chain and the XX model, we show that hydrodynamics captures the scaling behavior close to the light cone. On the other hand, our numerical analysis suggests that hydrodynamics fails in more general models, whenever a condition is not satisfied.Over the past few years, we are experiencing an increasing interest in the physics behind the nonequilibrium time evolution of inhomogeneous states. An example is the time evolution of two semi-infinite chains that are joined together after having been prepared in different equilibrium conditions [1,2]. This kind of settings allows one to investigate the transport properties of quantum many-body systems even if the system is isolated from the environment.The first analytic results in this context were obtained in noninteracting models . There, under the assumption of quasistationarity, a semiclassical picture applies where the information about the initial state is carried by free stable quasiparticles moving throughout the system. Similar results were obtained in the framework of conformal field theory and Luttinger liquid descriptions [25][26][27][28][29][30][31][32][33][34][35][36][37]. In the presence of interactions the situation was less clear [38][39][40][41][42][43][44][45][46][47], but, eventually, Refs [48,49] have shown that the continuity equations satisfied by the (quasi)local conserved quantities are sufficient to characterize the late-time behavior. The framework developed in Refs [48,49] is now known as generalized hydrodynamics [48], where "generalized" is used to emphasize that integrable models have infinitely many (quasi)local charges [50]. We will generally omit "generalized" and refer to the system of equations derived in [48,49] as first-order hydrodynamics, 1 st GHD, to emphasize that it is a system of first-order partial differential equations.Within 1 st GHD, it was possible to compute the profiles of local observables [48,49,[51][52][53][54][55][56][57], to conjecture an expression for the time evolution of the entanglement entropy [58], and to efficiently calculate Drude weights [59][60][61][62][63]. There are however fundamental questions that can not be addressed within 1 st GHD; diffusive transport [64][65][66][67][68][69] and large-tim...

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“…[89], it is simple to show that, if α > 0, K α (u, v) can be expressed in terms of the Airy kernel (11), as…”

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confidence: 99%

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

2017

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“…Notice that (3.1) is of the Hankel transform type. Earlier an expression of the Fourier transform type has been obtained in [26] which in the above variables can be written as …”

Section: Theoremmentioning

confidence: 99%

Integrals involving products of Airy functions, their derivatives and Bessel functions

Варламов¹

2010

Journal of Mathematical Analysis and Applications

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A new integral representation of the Hankel transform type is deduced for the functionThis formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function |Ai(z)| 2 with z ∈ C.

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“…The products of Airy functions can also be considered special functions in their own right [2,21]. Integral representations for double, triple and quartic products of Airy functions were the subject of a series of papers [3,4,16,[21][22][23]27]. Integral representations for Ai 2 (x) and Ai(x)Bi(x), where Ai(x) and Bi(x) are the Airy functions of the first and second kind, respectively, first appeared in the papers [3,4] …”

Section: Introductionmentioning

confidence: 99%

Fractional derivatives of products of Airy functions

Варламов¹

2008

Journal of Mathematical Analysis and Applications

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Fractional derivatives of the products of Airy functions are investigated, D α {Ai 2 (x)} and D α {Ai(x) × Bi(x)}, where Ai(x) and Bi(x) are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of D α {Ai(x)} and D α {Gi(x)}, where Gi(x) is the Scorer function. It is also proved that the Wronskian W(x) of the system of half integrals {D −1/2 Ai(x), D −1/2 Gi(x)} and its Hilbert transform W(x) = −H W(x) can be considered special functions in their own right since they are expressed in terms of Ai 2 (x) and Ai(x)Bi(x), respectively. Various integral relations are established. Integral representations for D α {Ai(x − a)Ai(x + a)} and its Hilbert transform −H D α {Ai(x − a)Ai(x + a)} are derived.

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An integral representation for the product of Airy functions

Cited by 27 publications

References 1 publication

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

Integrals involving products of Airy functions, their derivatives and Bessel functions

Fractional derivatives of products of Airy functions

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