By T. Aoki, H. Majima, Y. Takei, N. Tose

ISBN-10: 443173239X

ISBN-13: 9784431732396

This quantity comprises 23 articles on algebraic research of differential equations and similar issues, such a lot of that have been awarded 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 strong instruments which have been utilized to linear and non-linear differential equations in addition to many comparable fields equivalent to genuine and intricate research, indispensable transforms, spectral idea, inverse difficulties, integrable structures, and mathematical physics. The articles contained right here current many new effects and concepts, offering researchers and scholars with precious feedback and instructive tips for his or her paintings. 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 social gathering of Professor Kawai's sixtieth birthday as a token of deep appreciation of the $64000 contributions he has made to the sector. Introductory notes at the medical works of Professor Kawai also are included.

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**Additional info for Algebraic analysis of differential equations: from microlocal analysis to exponential asymptotics; Festschrift in honor Prof. Takahiro Kawai [on the occasion of his sixtieth birthday]**

**Sample text**

To describe the recipe, let us ﬁrst ﬁx the situation to be considered: Suppose that a Stokes curve γ1 of type (1,2) that emanates from a turning point τ1 intersects at a point ι with another Stokes curve γ2 of type (2,3) emanating from another turning point τ2 . In what follows τ1 and τ2 may be either virtual or traditional. (In case τ1 or τ2 is a simple turning point we need some 40 Takashi Aoki et al. ) Having the labeling of Fig. 3 in mind, a point x∗ that satisﬁes x∗ τ2 x∗ ξ1 dx = ξ2 dx + τ1 τ1 ξ3 dx (14) τ2 is a virtual turning point.

M. Nevins and K. V. Roberts: New Stokes’ line in WKB theory, J. Math. , 23 (1982), 988-1002. R. Courant and D. Hilbert: Methods of Mathematical Physics, II, Interscience, 1962. E. Delabaere, H. Dillinger and F. Pham: Exact semi-classical expansions for one dimensional quantum oscillators, J. Math. , 38 (1997), 61266184. L. , 127 (1971), 79183. N. Honda: Toward the complete description of the Stokes geometry, in prep. C. J. Howls, P. J. Langman and A. B. Olde Daalhuis: On the higher-order Stokes phenomenon, Proc.

Or “cognate”, singularities of ψB (x, y) coalesce. Actually a traditional turning point of a Schr¨ odinger operator P = d2 /dx2 − η 2 Q(x) is of this character: the ϕ of the equation P ϕ = 0 has Borel transform ϕB (x, y) of a WKB solution x√ two singularities s± = {(x, y); y = ± a Qdx} with Q(a) = 0, and they coalesce at (x, y) = (a, 0). Let us now raise the following question: In what sense are s+ and s− cognate? To answer this question, we have to understand the structure of singularities of ϕB (x, y).

### Algebraic analysis of differential equations: from microlocal analysis to exponential asymptotics; Festschrift in honor Prof. Takahiro Kawai [on the occasion of his sixtieth birthday] by T. Aoki, H. Majima, Y. Takei, N. Tose

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