Physical and mathematical aspects of Gamow states

Civitarese, O.

doi:10.1016/s0370-1573(04)00108-5

Cited by 45 publications

(102 citation statements)

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“…In next two sections, we detail two different methodologies for DSWs in nonlocal media: the TAQM [11,[56][57][58][59][60][61][62][63][64][65][66] and the wave turbulence theory [67,69]. Both theories also prove that DSWs are intrinsically irreversible.…”

Section: Spatial Dispersive Shock Waves In Nonlocal Kerr Nonlinearitymentioning

confidence: 99%

“…For many years, Whitham modulation and hydrodynamic approximation have predominated in solving the nonlocal NLSE [2,27]. However, hydrodynamic approximation cannot describe light propagating beyond the shock point, and two new techniques have been developed, one coming from nuclear physics, the time asymmetric quantum mechanics (TAQM) [11,[56][57][58][59][60][61][62][63][64][65][66], which models the nonlinear wave intrinsically irreversible propagation as a superposition of decaying resonances, and the wave turbulence theory [31,37,39,[67][68][69], which uses a statistical interpretation of nonlinear optics.…”

Section: Introductionmentioning

confidence: 99%

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Optical spatial shock waves in nonlocal nonlinear media

Marcucci

Pierangeli

Gentilini

et al. 2019

Advances in Physics: X

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Dispersive shock waves are fascinating phenomena occurring when nonlinearity overwhelms linear effects, such as dispersion and diffraction. Many features of shock waves are still under investigation, as the interplay with noninstantaneity in temporal pulses transmission and nonlocality in spatial beams propagation.Despite the rich and vast literature on nonlinear waves in optical Kerr media, spatial dispersive shock waves in nonlocal materials deserve further attention for their unconventional properties. Indeed, they have been investigated in colloidal matter, chemical physics and biophotonics, for sensing and control of extreme phenomena.Here we review the last developed theoretical models and recent optical experiments on spatial dispersive shock waves in nonlocal media. Moreover, we discuss observations in novel versatile materials relevant for soft matter and biology.

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Section: Spatial Dispersive Shock Waves In Nonlocal Kerr Nonlinearitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optical spatial shock waves in nonlocal nonlinear media

Marcucci

Pierangeli

Gentilini

et al. 2019

Advances in Physics: X

View full text Add to dashboard Cite

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“…2. Using Hardy functions on a half-plane [24][25][26], we may construct RHS that serve as a framework for an extension of ordinary quantum mechanics that accounts for time asymmetric quantum processes. One example of such processes is the quantum decay [27][28][29][30][31][32].…”

Section: Rigged Hilbert Spacesmentioning

confidence: 99%

Groups, Special Functions and Rigged Hilbert Spaces

Celeghini

Gadella

Olmo

2019

Axioms

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We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space, instead they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . As a matter of fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can be often continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We have investigated this formalism to various examples of interest in quantum mechanics. In particular, we have considered, SO(2) and functions on the unit circle, SU (2) and associated Laguerre functions, Weyl-Heisenberg group and Hermite functions, SO(3, 2) and spherical harmonics, su(1, 1) and Laguerre functions, su(2, 2) and algebraic Jacobi functions and, finally, su(1, 1)⊕su(1, 1) and Zernike functions on a circle.

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“…The action of F ∈ Ψ × on each vector ϕ ∈ Ψ gives a complex number often denoted as ϕ|F , following the Dirac notation. It is not our intention here to review properties and applications of RHS, the reader may visit for this purpose the extensive literature on the subject [5][6][7][8][9][10][11][12][13], but we would like to mention that recently they have received renewed attention, see [14][15][16][17][18][19], including their relations with partial inner products [20] and frames [21]. We limit us to remember that our primary purpose is to provide of mathematical sense to the Dirac formulation of quantum mechanics and its continuous basis of eigenvectors of observables.…”

Section: Introductionmentioning

confidence: 99%

Groups, Jacobi functions, and rigged Hilbert spaces

Celeghini

Gadella

Olmo

2020

Journal of Mathematical Physics

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This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations of their algebra of symmetry, that induce discrete and continuous bases coexisting on a rigged Hilbert space supporting the representation. Meaningful operators are shown to be continuous on the spaces of test vectors and its dual. Here, the chosen special functions, called "Algebraic Jacobi Functions" are related to the Jacobi polynomials and the Lie algebra is su(2, 2). These functions with m and q fixed, also exhibit a su(1, 1)-symmetry. Different discrete and continuous bases are introduced. An extension in the spirit of the associated Legendre polynomials and the spherical harmonics is presented introducing the "Jacobi Harmonics" that are a generalization of the spherical harmonics to the three-dimensional hypersphere S 3 .∞ j≥sup(|m|,|q|) |J m,q j (x) J m,q j (y)| ≡ ∞ j≥sup(|m|,|q|) J m,q j (x) J m,q j (y) = δ(x − y) .Note that the AJF have a direct relation with the Legendre functions:as can be obtained from [30] and [31] (p. 213).

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Physical and mathematical aspects of Gamow states

Cited by 45 publications

References 0 publications

Optical spatial shock waves in nonlocal nonlinear media

Optical spatial shock waves in nonlocal nonlinear media

Groups, Special Functions and Rigged Hilbert Spaces

Groups, Jacobi functions, and rigged Hilbert spaces

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