By Alexander O.E. Animalu

Intermediate Quantum idea of Crystalline Solids by way of Alexander O.E. Animalu

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**Example text**

2 shows that the geometric situation is basically the same as that observed for the Schrödinger equations, as far as x is confined to a neighborhood U of a simple turning point. To make full use of this we now posit the following Property [AC] as a guiding principle; it is a counterpart of Facts A and C observed for the Borel transformed WKB solutions of the Airy equation. Property [AC] The Borel transformed WKB solutions ϕl,B (x, y) (l = 1, 2, . . , without encountering the natural boundaries.

Note that a bidirectional binary tree of degree other than 2 always contains, by definition, a part of a new Stokes curve as its edges, and hence, a virtual turning point is indispensable in finding a bidirectional binary tree. As we want to explain the core part of the problem in a concise manner, we do not discuss in this article the procedure to find finitely many virtual turning points needed for the description of the Stokes geometry; that is, we start with the model of the Stokes geometry in the terminology of [H3].

II, Sect. , a maximally over-determined system, abbreviated as MOS, is more appealing to the intuition of the reader. 4) and then we can study the bicharacteristic flow in T ∗ C3(x,y) to define the notion of a virtual turning point. Similar holonomic systems with a large parameter η also appear [A] in analyzing the Shudo integral ··· exp ηS(q0 , q1 , . . 5) where n S(q0 , q1 , q2 , . . 7) with c being a real constant. The Shudo integral is a basic quantity in the study of the quantized Hénon map and Shudo [Sh] studied its analytic structure (for fixed q0 ) using the notion of virtual turning points and new Stokes curves.

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