Asymptotic and geometric quantization

Karasev, M. V.; Maslov, Victor Pavlovich

doi:10.1070/rm1984v039n06abeh003183

Cited by 64 publications

(38 citation statements)

References 114 publications

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“…Since p(z) is given by (112) nearz = 1, let us substitute it in the formula (19) for the scalar product and compute the asymptotics of the resulting integral. Let…”

Section: Theorem 4 the Function ω(Z Z) Attains Its Maximum Value Atzmentioning

confidence: 99%

“…After substituting the functions (163) in the formula (19) for the scalar product, we have an integral…”

Section: Lemma 16 Suppose the Conditionsmentioning

confidence: 99%

“…Then μ 1 and we can apply a quantum version of the averaging method [19], [20], [9] to the problem (4). The basic idea of this method is to find an invertible operator U and an operator S 1 + μS 2 such that…”

Section: Algebraic Averagingmentioning

confidence: 99%

“…Let P[m, n] denote the space of antiholomorphic polynomials over C of degree at most n − |m| − 1, with scalar product given by (19) (…”

Section: Coherent Transformationmentioning

confidence: 99%

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Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Pereskokov

2013

Trans. Moscow Math. Soc.

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Abstract. We investigate the second-order Zeeman effect in a magnetic field using irreducible representations of an algebra with the Karasev -Novikova quadratic commutation relations. To each such representation there corresponds a spectral cluster near the energy level of the unperturbed hydrogen atom. Using this model as an example, we describe a general method for constructing asymptotic solutions near the boundaries of spectral clusters based on a new integral representation. We also study the problem of computing quantum averages near the lower boundaries of clusters.

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“…Since p(z) is given by (112) nearz = 1, let us substitute it in the formula (19) for the scalar product and compute the asymptotics of the resulting integral. Let…”

Section: Theorem 4 the Function ω(Z Z) Attains Its Maximum Value Atzmentioning

confidence: 99%

“…After substituting the functions (163) in the formula (19) for the scalar product, we have an integral…”

Section: Lemma 16 Suppose the Conditionsmentioning

confidence: 99%

Section: Algebraic Averagingmentioning

confidence: 99%

“…Let P[m, n] denote the space of antiholomorphic polynomials over C of degree at most n − |m| − 1, with scalar product given by (19) (…”

Section: Coherent Transformationmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Pereskokov

2013

Trans. Moscow Math. Soc.

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“…Such an exact formula could be applied, for instance, to highly oscillating or singularly concentrated (as → 0) functions on T * M which are required to describe Schrödinger quantum dynamics or eigenfunction problems on M. On this topic we recall the asymptotic quantization theory [45] which allows this type of 'semiclassical' -dependence in its symbols and deals with symplectic manifolds of general type without having a global polarization. However for symplectic manifolds, ≺T to select in a unique way the quantization operation f →f (1.4) on a function space over T * M (Sect.…”

Section: Introductionmentioning

confidence: 99%

Cotangent bundle quantization: entangling of metric and magnetic field

Karasev

Osborn

2005

J. Phys. A: Math. Gen.

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For manifolds M of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space L 2 (T * M) and construct an irreducible representation of this algebra in L 2 (M). This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over T * M is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in M. The quantization of δ-functions induces a family of symplectic reflections in T * M and generates a magneto-geodesic connection Γ on T * M. This symplectic connection entangles, on the phase space level, the original affine structure on M and the magnetic field. In the classical approximation, the 2 -part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor.

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Algebras of Pseudodifferential Operators in L₂ Given by Smooth Measures on Hilbert Spaces

Albeverio

Daletskh

1998

Mathematische Nachrichten

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Asymptotic and geometric quantization

Cited by 64 publications

References 114 publications

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Cotangent bundle quantization: entangling of metric and magnetic field

Algebras of Pseudodifferential Operators in L₂ Given by Smooth Measures on Hilbert Spaces

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Asymptotic and geometric quantization

Cited by 64 publications

References 114 publications

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Cotangent bundle quantization: entangling of metric and magnetic field

Algebras of Pseudodifferential Operators in L2 Given by Smooth Measures on Hilbert Spaces

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Product

Resources

About

Algebras of Pseudodifferential Operators in L₂ Given by Smooth Measures on Hilbert Spaces