Prog. Theor. Phys. Vol. 122 No. 6 (2009) pp. 1311-1346

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Overlooked Branch Cut in Steepest Descent Method

— Switching Line and Atomic Domain —

Tadanori Hyouguchi,^1,2 Ryohei Seto^3,4 and Satoshi Adachi⁵

¹Research Organization of Science and Engineering, Ritsumeikan University, Kusatsu 525-8577, Japan
²Department of Applied Mathematics and Informatics, Ryukoku University, Otsu 520-2194, Japan
³Laboratoire de Génie Chimique, Bât 2R1 31062 Toulouse Cedex 9, Université Paul Sabatier, France
⁴Department of Physical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan
⁵Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

(Received September 9, 2009)

Abstract:

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.

Subject Index : 013, 064
URL : http://ptp.ipap.jp/link?PTP/122/1311/
DOI : 10.1143/PTP.122.1311

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Citing Article(s) :

Progress of Theoretical Physics Vol. 122 No. 6 (2009) pp. 1347-1376 :
Overlooked Degree of Freedom in Steepest Descent Method

Tadanori Hyouguchi, Ryohei Seto and Satoshi Adachi