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By Nanny Fröman, Per Olof Fröman

This e-book covers the most effective approximation tools for the theoretical research and resolution of difficulties in theoretical physics and utilized arithmetic. the strategy will be utilized to any box concerning moment order usual differential equations. it's written with useful wishes in brain, with 50 solved difficulties overlaying a vast diversity of matters and making transparent which suggestions and result of the overall concept are wanted in each one case.

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Physical Problems Solved by the Phase-Integral Method

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O. Fr¨oman (1980) in which the important difference between the two types of approximations was emphasized. 3a,b), were derived in a systematic way by N. Fr¨oman (1966d). Using the recurrence formula for the WKB series, she showed that the usual WKB series, with infinitely many terms retained, can be transformed so that the sum of the terms of odd order in the expansion parameter can be simply expressed in terms of the sum of the terms of even order, and she thus obtained a formal solution of the time-independent Schr¨odinger equation containing only the even-order terms.

20a,b) have important applications. 5 Stokes and anti-Stokes lines We have seen above that, when tracing a solution in the complex z-plane, one must know how the monotonicity of the absolute value of exp[iw(z)] changes along the path. 21a) dw = q(z)dz is real along an anti-Stokes line. 21b) Thus, the Stokes lines are lines along which the absolute value of exp[iw(z)] increases or decreases most rapidly, while the anti-Stokes lines are level lines for constant absolute values of exp[iw(z)]. 4. 6.

24b) where and 32 2 Description of the phase-integral method as z tends to +∞ along the real z-axis. These exact solutions represent outgoing and incoming waves, respectively, as z tends to +∞ along the real z-axis. We now assume that the potential V (z) has such properties that the exact functions χ+ (z) and χ− (z) tend to k −1/2 and their derivatives tend to zero not only as z → +∞ along the real z-axis, but also as the absolute value of z tends to infinity when arg z lies in an interval of finite size.

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