By S. Yu. Slavyanov

ISBN-10: 0821805363

ISBN-13: 9780821805367

This e-book is dedicated to asymptotic research of strategies of moment order traditional differential equations with a small parameter. the most emphasis is on a number of positive schemes of acquiring asymptotic ideas, their benefits and disadvantages, and particular computations. the writer offers a whole review of the kingdom of the idea and likewise concentrates on a few lesser recognized elements and difficulties, particularly the issues within which exponentially small phrases could be taken into consideration or the research of equations with shut transition issues. Such purposes because the derivation of the formulation for the quasiclassical quantizations, spectrum splitting in a symmetrical capability, etc., are thought of.

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**Extra info for Asymptotic solutions of the one-dimensional Schrödinger equation**

**Example text**

Zo. 10); that is, 11±(z) = z- 1/ 4 exp ( ± 2 · z 312/3). 22Y are defined by taking arithmetic values for positive arguments; 6(z) = 5/(16z2). Set c1 = 0, c2 = 1 and let the integration be from +oo to z. This will give us one of the solutions of the Airy equations, y- (z). As we will see, up to a multiplicative factor this solution coincides with Ai(z). 23) a(z) = 1 + : 2 12 { 1 - exp [4(z312 - ( 312)/3] }a(() d(. ()() We now change the variables as t = 2z3 / 2/3, s = 2(3 12/3 and define the function · a(t) = a(z).

Consider the contour integral I = I e-t2 /2+tzrv-1 dt, l defined for any value of a complex parameter v. The contour l, shown in Figure 7, encircles the cut ( -oo, OJ. If we fix the branch of the integrand by the condition arg(rv- 1 ) = 0 for argt = O, we can rewrite the above integral for Rev< 0 as J J 00 I= e-i'll"v e-t2/2-tzt-v-ldt _ ei'll"v 0 00 e-t 212-tzrv-ldt. 3. 3) I e-t2/2-tzrv-1 dt. 2) to the right halfplane of v. 2), and the extension to the rest of the complex v-plane will be carried out by the analytic continuation.

21) y(z) = c111+(z) :f- c211-(z) + J K(z,()6(()y(() d(, zo where 1 . K(z, () = W( 11+, 11 _) (11+(z)11-(() -11-(z)11+(()), and W (11+, 11-) is the Wronskian of 11+, 11-. It can be calculated explicitly: W(11+,11-) =11+11- 1 -11-11+ 1 = -2. Here z0 is either a fixed finite point or oo. 21) is carried out along a finite contour connecting the points zo and z. zo. 10); that is, 11±(z) = z- 1/ 4 exp ( ± 2 · z 312/3). 22Y are defined by taking arithmetic values for positive arguments; 6(z) = 5/(16z2).

### Asymptotic solutions of the one-dimensional Schrödinger equation by S. Yu. Slavyanov

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