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Asymptotic Analysis 23 (2000) 91–109 91 IOS Press Complex WKB method for 3-level scattering systems Alain Joye a and Charles-Edouard Pfister b a Institut Fourier, Université de Grenoble-1, BP 74, F-38402 Saint-Martin d'Hères Cedex, France E-mail: b Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland E-mail: Abstract. In this note the S-matrix naturally associated with a singularly perturbed three-dimensional system of linear differ- ential equations without turning point on the real axis is considered. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circumstances. In particular, we show how to compute asymptotically all exponentially small off-diagonal elements of the corresponding S-matrix. Keywords: Singular perturbations, semiclassical analysis, adiabatic approximations, exponential asymptotics, n-level S-matrix, turning point theory. 1. Introduction We consider the computation of the leading term of exponentially small elements of the S-matrix natu- rally associated with singularly perturbed 3-dimensional systems of linear ordinary differential equations without turning points on the real axis by means of the complex WKB method. Several progresses have been made during the last few years on general aspects of this method in several directions, such as the improvement of the asymptotics it yields [11] or its application to systems of ODE of higher dimen-
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Asymptotic Analysis 23 (2000) 91–109 91 IOS Press Complex WKB method for 3-level scattering systems Alain Joye a and Charles-Edouard Pfister b a Institut Fourier, Université de Grenoble-1, BP 74, F-38402 Saint-Martin d'Hères Cedex, France E-mail: b Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland E-mail: Abstract. In this note the S-matrix naturally associated with a singularly perturbed three-dimensional system of linear differ- ential equations without turning point on the real axis is considered. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circumstances. In particular, we show how to compute asymptotically all exponentially small off-diagonal elements of the corresponding S-matrix. Keywords: Singular perturbations, semiclassical analysis, adiabatic approximations, exponential asymptotics, n-level S-matrix, turning point theory. 1. Introduction We consider the computation of the leading term of exponentially small elements of the S-matrix natu- rally associated with singularly perturbed 3-dimensional systems of linear ordinary differential equations without turning points on the real axis by means of the complex WKB method. Several progresses have been made during the last few years on general aspects of this method in several directions, such as the improvement of the asymptotics it yields [11] or its application to systems of ODE of higher dimen-
- incorrect results
- when ?
- equations without turning
- all eigenvalues
- global asymptotic theory
- called superasymptotic
- differ- ential equations
Asymptotic Analysis 23 (2000) 91109 IOS Press
91
Complex WKB method for 3-level scattering systems
Alain Joye a and Charles-Edouard Pster b a Institut Fourier, Université de Grenoble-1, BP 74, F-38402 Saint-Martin d'Hères Cedex, France E-mail: joye@fourier.ujf-grenoble.fr b Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland E-mail: cpster@eldp.ep.ch Abstract. In this note the S -matrix naturally associated with a singularly perturbed three-dimensional system of linear differ-ential equations without turning point on the real axis is considered. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circumstances. In particular, we show how to compute asymptotically all exponentially small off-diagonal elements of the corresponding S -matrix. Keywords: Singular perturbations, semiclassical analysis, adiabatic approximations, exponential asymptotics, n -level S -matrix, turning point theory.
1. Introduction We consider the computation of the leading term of exponentially small elements of the S -matrix natu-rally associated with singularly perturbed 3-dimensional systems of linear ordinary differential equations without turning points on the real axis by means of the complex WKB method. Several progresses have been made during the last few years on general aspects of this method in several directions, such as the improvement of the asymptotics it yields [11] or its application to systems of ODE of higher dimen-sion than two [7,3]. However, it is well known [2], that in general the S -matrix cannot be completely determined asymptotically for systems of dimension higher than 2. In this note, we present a model whose study illustrates the fact that the complex WKB method can actually give results for specic cases going beyond those proven in [7] or [3] for “generic three-dimensional systems. Indeed, for this model the whole S -matrix is computed asymptotically. Moreover, and this is the main point of this study, this is true for a whole class of systems we describe at the end of the paper. Before introducing our model, let us mention that the complex WKB theory has a very long history which can be retraced in the classics [4,15,14,1] for example. More recent developpements as well as studies of non-generic situations can be found in [10,11,13,7,3] and references therein. The reader is directed to this non-exhaustive list for an historic point of view and precise references on the general aspects of the theory. We now dene our model and then explain in more details the strategy we will follow to determine the corresponding S -matrix. Consider the following system in the singular limit ε → 0 iε 0 ( t ) = H ( t , ) ( t ), t ∈ R , ε → 0, (1.1)
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