The higher-order heat-type equations via signed Lévy stable and generalized Airy functions

Górska, K.; Horzela, A.; Penson, K. A.; Dattoli, G.

doi:10.1088/1751-8113/46/42/425001

Cited by 26 publications

(38 citation statements)

References 60 publications

(211 reference statements)

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“…Explicit expressions. Similar functions were defined in [82] as Green functions of higher order diffusion equations, and some of their properties were studied there. It was shown that they admit explicit expressions in terms of hypergeometric functions.…”

Section: Differential Equationmentioning

confidence: 99%

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Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

2018

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We compute the joint statistics of the momenta p_{i} of N noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one p_{max}. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, p_{max}, as x_{max}, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e., for "flat traps" near their minimum, with V(x)∼x^{2n} and n>1. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the nth member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

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Section: Differential Equationmentioning

confidence: 99%

“…It was shown that they admit explicit expressions in terms of hypergeometric functions. Here we give only the case n = 2, where it reads (correcting for some misprints in [82])…”

Section: Differential Equationmentioning

confidence: 99%

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

2018

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“…In such a lattice framework only n F n−1 hypergeometric functions [73] with regular singularities occur. Of course irregular singularities can also occur in physics [82,83,84], in particular in the scaling limit of lattice models [85,86] (modified Bessel functions, etc ...).…”

Section: Functions Deduced From the Goursat Identitymentioning

confidence: 99%

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

110

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Abstract.We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same 2 F 1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus 2 F 1 hypergeometric function example. We then focus on identities relating the same 2 F 1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale's paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to 3 F 2 , hypergeometric functions, and show that one just reduces to the previous 2 F 1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a 2 F 1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or 2 F 1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

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“…in line with (94). As a conclusion, we mention that the result (19) as well as its normalized form (35) can be considered as the analytic continuation of those for the equation [78,79]…”

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confidence: 85%

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Torre

Lattanzi²,

Levi

2016

Annalen der Physik

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A detailed study of the spinless (1+1)D free-particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter "flow" in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested. +∞ −∞ e −i p 2 2 m t e i px ψ 0 ( p)dp. (12) As we know, in analogy with (9), the evolution under the Schrödinger equation amounts in the p-domain to an energyrelated phase-factor, and hence to a simple quadratic-phase

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The higher-order heat-type equations via signed Lévy stable and generalized Airy functions

Cited by 26 publications

References 60 publications

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

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