WKB analysis for high-order ordinary differencial equations with a large parameter

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The purpose of this research was to establish the exact WKB analysis for ordinary differential equations of higher-order with a large parameter. Regarding local theory, we have achieved this purpose. The results we have obtained are as follows:(1) We established a new method of describing Stokes geometry for differential equations of higher-order with a large parameter. This method is called the exact steepest descent method. By using this method, we can find complete Stokes geometry for linear ordinary differential equations of arbitrary order with quadratic coefficients. This is a natural generalization of the steepest descent method.(2) We consider linear ordinary differential equation of infinite order with a large parameter satisfying the following condition: Regarding the large parameter as a differential operator with respect to the variable of the Borel plane, they are microdifferential operators of order 0 and they do not contain that variable. The category of such operators c … More ontains linear ordinary differential operators of arbitrary order with the large parameter. For such an operator, we have defined the notions of WKB solutions, turning points and Stokes curves. We have also introduced the notion of simplicity of turning points. These notions are natural extension of that for finite-order case,(3) Such an equation in the class mentioned above may admit infinitely many phases. After fixing one of it, we have constructed the WKB solution of the difierential equation. This construction is applicable to linear ordinary differential equations of arbitrary order.(4) We have established local decomposition theorem near turning points. If we consider a simple turning point of a differential equation of infinite order, we can decompose the relevant differential operator into the product of two operators; one is invertible and another is of second order whose turning point is exactly the same as the simple turning point. Moreover, the phase of the second order operator is the coincident with the original phase. This implies that an infinite-order differential equation is reduced to a second-order equation near a simple turning point. Thus we have obtained local connection formulas of infinite-order equations near simple turning points. Less