Overlooked Branch Cut in Steepest Descent Method : Switching Line and Atomic Domain(General and Mathamatical Physics)

概要

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One type of branch cut has. been overlooked in the steepest descent method. This type of branch cut, which we call the switching line, is inevitable in this method if one wants to obtain the asymptotic evaluation of a target integral I(x) globally for any value of control parameter x. The appearance of switching line on the space of control parameter can be viewed as one example of Berry's adiabatic phase. The recognition of the switching line leads us to yield a useful recipe of steepest descent method. Within the recipe, the whole space of control parameter is divided into multiple domains by the three types of boundary line (saddle connection line, ordinary branch cut and switching line). Each resultant domain obtained from this division is termed the atomic domain. The topological feature of stationary phase paths over each atomic domain can be reduced to what we call the connectivity graph. With use of the connectivity graph, the final asymptotic expression of a given integral I(x) for any value of control parameter x can be presented very clearly without any ambiguity.
2009-12-25