By Takashi Aoki, Hideyuki Majima, Yoshitsugu Takei, Nobuyuki Tose (eds.)
This quantity includes 23 articles on algebraic research of differential equations and comparable themes, such a lot of which have been provided as papers on the overseas convention "Algebraic research of Differential Equations – from Microlocal research to Exponential Asymptotics" at Kyoto college in 2005.
Microlocal research and exponential asymptotics are in detail hooked up and supply robust instruments which have been utilized to linear and non-linear differential equations in addition to many comparable fields corresponding to actual and complicated research, imperative transforms, spectral conception, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and ideas.
This quantity is devoted to Professor Takahiro Kawai, who's one of many creators of microlocal research and who brought the means of microlocal research into exponential asymptotics. This commitment is made at the get together of Professor Kawai's sixtieth birthday as a token of deep appreciation of the $64000 contributions he has made to the sphere. Introductory notes at the clinical works of Professor Kawai also are included.
Read Online or Download Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai PDF
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Extra info for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai
Press, 2000, pp. 245–260. : Toward the exact WKB analysis for higher-order Painlev´e equations – The case of Noumi-Yamada Systems –, Publ. , 40 709–730 (2004) Ghost busting: Making sense of non-Hermitian Hamiltonians Carl M. Bender Physics Department, Washington University, St. edu Summary. The Lee model is an elementary quantum ﬁeld theory in which mass, wave-function, and charge renormalization can be performed exactly. Early studies of this model in the 1950’s found that there is a critical value of g 2 , the square of the renormalized coupling constant, above which g02 , the square of the unrenormalized coupling constant, is negative.
VI) (, ). Making use of these geometric features, Kawai and I proved in  (see also its announcement paper ) that we can construct an invertible transformation which brings (SLJ ) to (SLI ) on a neighborhood of the above mentioned Stokes curve γ and further that it describes an equivalence between a 0-parameter solution of (PJ ) and that of (PI ) near a simple turning point of (PJ ). Later in  and  we extended this result to a “2-parameter solution” (or “instanton-type solution”) of (PJ ) of the form j+1 λJ (t, η) = λ0 (t) + η −1/2 η −j/2 j≥0 (j) a2k−j−1 (t) exp((2k − j − 1)φ(t)η) , k=0 (8) which was constructed through multiple-scale analysis in .
K. Proof. Let mj denote the degree of fj (j = 0, 1, . . , k) and set μ = max deg(Rj fj ). j We prove the lemma by induction on μ. The lemma trivially holds if μ < 0. Suppose that the lemma is proved for μ − 1. Taking σμ of the relation k Rj fj = 0, (10) j=0 we have k σμ−mj (Rj )σ(fj ) = 0. j=0 Since the sequence σ(f0 ), σ(f1 ), . . , σ(fk ) is a tame regular sequence, there exist hij ∈ R satisfying deg(hij ) = μ − mi − mj , hij = −hji and σμ−mi (Ri ) = hij σ(fj ). j Now we set ˜ i = Ri − R hij fj .