Lagrangian Manifolds and the Maslov Operator

Mishchenko, A. S.; Shatalov, V. E.; Sternin, B. Yu.

doi:10.1007/978-3-642-61259-6

Cited by 76 publications

(98 citation statements)

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“…We will work with the A-manifold, since the B-case is similar. For modern treatments of WKB theory, see for example Martinez (2002) or Mishchenko et al (1990). The variables x are half of the coordinates (x, p) on P ; we denote "x-space" by Q = R N , which abstractly is best seen as the quotient space when P is divided by the foliation into vertical Lagrangian planes, x = const.…”

Section: Densities and Amplitude Determinantsmentioning

confidence: 99%

Semiclassical mechanics of the Wigner 6j-symbol

Aquilanti¹,

Haggard²,

Hedeman³

et al. 2012

J. Phys. A: Math. Theor.

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Abstract. The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the PonzanoRegge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.

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Section: Densities and Amplitude Determinantsmentioning

confidence: 99%

Semiclassical mechanics of the Wigner 6j-symbol

Aquilanti¹,

Haggard²,

Hedeman³

et al. 2012

J. Phys. A: Math. Theor.

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“…Any linear subspace l ⊂ R 2n whose vectors are pairwise isotropic satisfies dim l ≤ n, since l is contained in the Euclidean subspace orthogonal to J·l = {J z | z ∈ l}. An n-dimensional linear subspace l ⊂ R 2n is a (real ) Lagrange plane if z T Jw = 0 for all z and w in l. The space L R of all real Lagrange planes of R 2n is a compact orientable manifold of dimension n(n + 1)/2: see Matsushima [21] and Mishchenko et al [22]. An element l of L R can be represented by a 2n × n real matrix…”

Section: Preliminariesmentioning

confidence: 99%

Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability

et al. 2015

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Abstract. In this paper the dissipativity of a family of linear-quadratic control processes is studied. The application of the Pontryagin Maximum Principle to this problem gives rise to a family of linear Hamiltonian systems for which the existence of an exponential dichotomy is assumed, but no condition of controllability is imposed. As a consequence, some of the systems of this family could be abnormal. Sufficient conditions for the dissipativity of the processes are provided assuming the existence of global positive solutions of the Riccati equation induced by the family of linear Hamiltonian systems or by a convenient disconjugate perturbation of it.

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“…The quantization condition (e.g., see [10,12]) is satisfied for L g (i.e., the Maslov index is zero on L g ).…”

Section: Quantized Canonical Transformationsmentioning

confidence: 99%