An integrable (classical and quantum) four-wave mixing Hamiltonian system

Odzijewicz, Anatol; Wawreniuk, E.

doi:10.1063/5.0006887

Cited by 4 publications

(3 citation statements)

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“…We omit here the subcase when M is a compact Riemann surface, since then the Hilbert space H postulated in Definition 4.1. has finite dimension, which makes the theory less interesting from a mathematical point of view, but not necessarily from a physical one, e.g. see [14,[30][31][32].…”

Section: Quantization Of Holomorphic Flows On Non-compact Riemann Surfacesmentioning

confidence: 99%

“…One can find a large class of systems quantizable by the coherent state method in optics [14,16,38], where one usually considers a finite number of modes of an electromagnetic field self-interacting through a nonlinear medium [11,33,39]. In the papers [30][31][32] the classical and quantum reduction procedures were applied to the system of nonlinearly coupled harmonic oscillators (modes) which leads to quantization of the Hamiltonian systems on circularly symmetric surfaces called Kummer shapes [13,30]. This is a case to which one can apply the results obtained in Sect.…”

Section: Remarks About Physical Applicationsmentioning

confidence: 99%

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Some Aspects of Positive Kernel Method of Quantization

Odzijewicz

Horowski

2021

Commun. Math. Phys.

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We discuss various aspects of the positive kernel method of quantization of the one-parameter groups $$\tau _t \in \text{ Aut }(P,\vartheta )$$ τ t ∈ Aut ( P , ϑ ) of automorphisms of a G-principal bundle $$P(G,\pi ,M)$$ P ( G , π , M ) with a fixed connection form $$\vartheta $$ ϑ on its total space P. We show that the generator $${\hat{F}}$$ F ^ of the unitary flow $$U_t = e^{it {\hat{F}}}$$ U t = e i t F ^ being the quantization of $$\tau _t $$ τ t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when $$G=\hbox {GL}(N,{\mathbb {C}})$$ G = GL ( N , C ) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $$\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )$$ τ t hol ∈ Aut ( P , ϑ ) . For the above case, we present the integral decompositions of the positive kernels on $$P\times P$$ P × P invariant with respect to the flows $$\tau _t^{\mathrm{hol}}$$ τ t hol in terms of the spectral measure of $${\hat{F}}$$ F ^ . These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on $${\mathbb {C}} \times {\mathbb {C}}$$ C × C invariant with respect to the one-parameter groups of translations of complex plane.

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Section: Quantization Of Holomorphic Flows On Non-compact Riemann Surfacesmentioning

confidence: 99%

Section: Remarks About Physical Applicationsmentioning

confidence: 99%

Some Aspects of Positive Kernel Method of Quantization

Odzijewicz

Horowski

2021

Commun. Math. Phys.

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“…as a four-wave mixing Hamiltonian. Let us mention that the non-linear four-wave mixing processes arrise in optics, mechanics, solid body physics and information theory, see [6,15,19]. Usually, the considered models of four waves systems are solved by numerical or approximative methods.…”

Section: Proof (I)mentioning

confidence: 99%

Integrable Hamiltonian systems on the symplectic realizations of $\textbf{e}(3)^*$

Odzijewicz,

Wawreniuk

2021

Preprint

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The phase space of a gyrostat with a fixed point and a heavy top is the Lie-Poisson space e(3) * ∼ = R 3 × R 3 dual to the Lie algebra e(3) of Euclidean group E(3). One has three naturally distinguished Poisson submanifolds of e(3) * : (i) the dense open submanifold R 3 × Ṙ3 ⊂ e(3) * which consists of all 4-dimensional symplectic leaves ( Γ 2 > 0 3) * and ν < 0, µ are some fixed real parameters. Basing on the U (2, 2)-invariant symplectic structure of Penrose twistor space we find full and complete E(3)-equivariant symplectic realizations of these Poisson submanifolds which are 8-dimensional for (i) and 6-dimensional for (ii) and (iii). As a consequence of the above Hamiltonian systems on e(3) * lift to the ones on the above symplectic realizations. In such a way after lifting integrable cases of gyrostat with a fixed point, as well as of heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.

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Symplectic Realizations of e(3)∗

Wawreniuk

2023

Trends in Mathematics

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An integrable (classical and quantum) four-wave mixing Hamiltonian system

Cited by 4 publications

References 28 publications

Some Aspects of Positive Kernel Method of Quantization

Some Aspects of Positive Kernel Method of Quantization

Integrable Hamiltonian systems on the symplectic realizations of $\textbf{e}(3)^*$

Symplectic Realizations of e(3)∗

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