Asymptotic Methods in Analysis

Fedoryuk, M. V.

doi:10.1007/978-3-642-61310-4_2

Cited by 97 publications

(88 citation statements)

References 5 publications

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“…These problems are tackled with success by means of pseudodifferential operator techniques and/or BKW methods that provide asymptotic solutions up to errors of order O(ε m ), where ε is the ratio of length scales, and m depends on the peculiarities of the problem. See the monographs [9,10,11,17,31,37], for example.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

Joye

Marx

2006

Comm Pure Appl Math

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We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime. We assume the matrix-valued coefficients are analytic in the space variable, and we further suppose that the corresponding dispersion relation admits real-valued modes only with one-dimensional polarization subspaces. Hence a BKW-type analysis of the solutions is possible. We typically consider time-dependent solutions to the PDE that are carried asymptotically in the past and as x → −∞ along one mode only and determine the piece of the solution that is carried for x → +∞ along some other mode in the future. Because of the assumed nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau-Zener mechanism. We completely elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing of finite width takes place between the involved modes.

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Section: Introductionmentioning

confidence: 99%

Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

Joye

Marx

2006

Comm Pure Appl Math

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“…Furthermore, the results we need, while undeniably implicit in these works, are not always explicitly stated, or easy to find and identify. In some cases, proofs are absent as well [AGZV88;Fed89].…”

Section: Further Topicsmentioning

confidence: 99%

Asymptotic expansions of oscillatory integrals with complex phase

Pemantle¹

2010

Algorithmic Probability and Combinatorics

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Abstract. We consider saddle point integrals in d variables whose phase functions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in complex d-space. This work is motivated by applications to asymptotic enumeration.

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“…Let us consider now regularized oscillatory integral Principle [25]- [26])Let Ω be a domain inℝ ௗ× and ݂ ∈ ‫ܥ‬ ∞ ሺΩ) be a smooth functionof compact support, ܵ ఌ ∈ ‫ܥ‬ ∞ ሺΩ), ߝ ∈ ሺ0,1ሿbe a real valued smooth function withoutstationary points in suppሺ݂), i.e.߲ ௫ ܵ ఌ ሺ‫)ݔ‬ ≠ 0 for ‫ݔ‬ ∈ Ω. Let ‫ܮ‬be a differential operator…”

Section: Introductionmentioning

confidence: 99%