By V.P. Maslov

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**Extra resources for Asymptotics of Operator and Pseudo-Differential Equations (Monographs in Contemporary Mathematics)**

**Example text**

We consider the equation (cf. Section 1:P): a2u a2u ax2+ (1 +b(x))a2u = 0, ult=0 s 0, utjt'0 - uo(x), (1) at2 where b(x) E Co(al). Intrcduce the operators Al =- iX A2=x, B=x. 3, where A(x) ix1 + x2}1/2, wl = xl/A(x) w2 - x2/A(x). Thus, the symbol G(t,xl,x2'a) of the resolving operator for the Cauchy problem (1) is sought in the form of an asymptotic expansion with respect to powers of A(xl,x2) {xl + xJy}'". The fact that such expansion gives the mixed smoothnessgrowth-at-infinity asymptotics follows from the estimates proved in Chapter 2.

First we note that (6) implies commutability of Proof of Lemma 2. the family R1. u, so that Ker Ru C Ker RA, and Ruu - RA(I + (u - A)RV)u, so that ImR,, C ImRA. ,. Conversely RI - (A - A)-1, then clearly Ker RA - (0}, since (A - A)R1 - I. let N = M. Define the operator Al with the domain R by the formula (7). Then clearly AA is closed (since RA is), (AI - AA)RA = 1, RA(AI - AA) - IIR (the restriction on R of identity operator). It remains to prove that AA We have does not depend on A. Let x E R, then x = RAv for some v E X.

The main novelty is that the classes [pdq] and one-dimensional index are not defined separately. The application of this developed apparatus to asymptotic problems begins from the papers 135] and 133]. 28 The notion of the sympiectic manifold of the Poisson algebra together with the notion of theasymptotic group algebra was introduced in the paper (36). Also the formulae for symbols of regular representation operators was announced in this paper (the complete proof is contained in 1341). Thus all the main topologic and geometric notions used in Chapters 3 and 4 were recently developed in the cited papers for the case of small parameter asymptotics.