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Download e-book for iPad: Asymptotic solutions of the one-dimensional Schrödinger by S. Yu. Slavyanov

By S. Yu. Slavyanov

ISBN-10: 0821805363

ISBN-13: 9780821805367

This publication is dedicated to asymptotic research of recommendations of moment order usual differential equations with a small parameter. the most emphasis is on a number of confident schemes of acquiring asymptotic strategies, their merits and downsides, and particular computations. the writer provides an entire review of the nation of the idea and likewise concentrates on a few lesser recognized features and difficulties, particularly the issues within which exponentially small phrases may be taken under consideration or the research of equations with shut transition issues. Such functions because the derivation of the formulation for the quasiclassical quantizations, spectrum splitting in a symmetrical capability, etc., are thought of.

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Additional resources for Asymptotic solutions of the one-dimensional Schrödinger equation

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A, c, z) and relations between them. 1) zw"(z) + (c - z)w'(z) - aw(z) = 0 is called the confluent hypergeometric equation. It has two singularities, regular at z = 0 and irregular at z = oo. 1) with some values of a and c by changing variables and the unknown function. Therefore, all the properties of the corresponding functions could be derived from the formulas presented in this section. However, such a method would be too tedious. z + .. z. 1). The function

28) - :z:2/4 -v-1 ( Vv(x) - e x 1+ ~ r(2k- v) -2k) t:i. 22k k! 3. PARABOLIC CYLINDER FUNCTIONS AND THEIR ASYMPTOTICS 29 e 3. The following notations are used: ( = z 2 /4, = x 2 /4, ao = 1, bo = 1, bk(11) = ak(-11-l), ak+i(11) = (k2" + v(~k 1 >)ak(11) r(2k-v) or ak (11 ) = k! 2a,. r(-v). /2ir 2-v+l(-(v+l)/2e< r(-11) x ( 1+ D,,(z) 2"ei11'v/2(v/2ef. f2ir 2-v-le-(v+l)/2e-F. 2r(-11) x (1 + 2" COS11"11e-F. ( 1 + z=-x f ~(-1)kbk(11)e-k) (-l)kak(11)e-k) k=l + rY::) 2-v-lef. ( 1 + ~bk(11)e-k) We see that on the positive real axis the asymptotics of the function V,,(x) contains only the dominant part.

Fii. {IZ E: > 0. 14 I. COMPARISON FUNCTIONS Ret + FIGURE 4 4. The Stokes phenomenon for the Airy equation. C1 (shaded) approaches the second saddle point and at largzl = 27r/3 passes through both saddle points (see Figure 5(a)). C1, being drawn along the steepest descent lines, breaks into two parts: the contour li passing through the saddle point t 1 and the contour l 2 passing through the saddle point t2 (see Figure 5(b)). To obtain the asymptotic expansion for Ai(z) it is now necessary to take into account the contribution of both saddle points.

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

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