An Introduction to Phase-Integral Methods

By John Heading

The phase-integral method in mathematics, also known as the Wentzel-Kramers-Brillouin (WKB) method, is the focus of this introductory treatment. Author John Heading successfully steers a course between simplistic and rigorous approaches to provide a concise overview for advanced undergraduates and graduate students in mathematics and physics.
Since the number of applications is vast, the text considers only a brief selection of topics and emphasizes the method itself rather than detailed applications. The process, once derived, is shown to be one of essential simplicity that involves merely the application of certain well-defined rules. Starting with a historical survey of the problem and its solutions, subjects include the Stokes phenomenon, one and two transition points, and applications to physical problems. An appendix and bibliography conclude the text.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 3, 2013
• ISBN: 9780486316291

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Preface

He is a bold writer who puts his hand to the production of the first text (albeit a small one) devoted exclusively to the subject of phase-integral methods, since in such an exacting subject as mathematics it is surprising that the development of this technique over the last fifty years has been the occasion of so much error, criticism and dispute. Moreover, the treatment of the subject in the literature ranges from the ridiculously simple void of all rigour to the most sophisticated, the former hardly deserving mention and the latter not forming part of what is commonly known as the W.K.B.J. method. Hence the author has picked his way between the two extremes in order to produce what he hopes will prove to be a readable and helpful account of the method, since its name is known to many but its actual technique is known to but few. Moreover, since the number of applications is vast, only a brief selection of topics is considered, concentration being focused upon the method itself rather than upon detailed applications; it is sufficient to state that the author hopes to produce a more comprehensive mathematical account elsewhere at a later date.

The text has been developed to show that the process, once derived, is one of essential simplicity, being merely the application of certain well-defined rules, although an element of controversy still cannot be ruled out. To this end, the author has not hesitated to introduce his own notation consistently to denote the W.K.B.J. solutions, thereby to avoid the usual conglomeration of definite integrals within exponentials that habitually characterises the printed page dealing with phase-integral methods. Finally, the author wishes to thank the Editor's referee, Dr. K. G. Budden of the Cavendish Laboratory, Cambridge, for many helpful comments and corrections.

J. H.

University of Southampton

August 1961

CHAPTER I

Historical Survey of the Problem

1.1 Introductory remarks

A linear differential equation of the first order

w′ + p(x) w = 0

(where a prime is used for different with respect to the independent variable) possesses an immediate solution in the form

w = exp [− ∫ p(x) dx].

On the other hand, an equation of the second order with variable coefficients such as

w˝ + p(x) w′ + q(x) w = 0

possesses no such simple general solution. Since many problems that arise in theoretical physics lead to differential equations of this type, attempts must be made to produce satisfactory methods of solution.

If p and q are specially chosen, the various standard transcendental differential equations are produced. For example, if p = 1/ x and q = 1 − v²/ x², we obtain Bessel's equation of order v; if p = 0 and q = − x, we obtain the Airy equation. These special cases are singled out because exact solutions are readily obtainable in terms of contour integrals, from which the power-series solutions in ascending powers of x and the asymptotic solutions in descending powers of large x may be obtained.

In all but standard cases, contour-integral solutions can no longer be found, and similarly the explicit forms of the coefficients occurring in the power-series and asymptotic solutions cannot be written down. Under these circumstances, approximations are necessary, and phase-integral methods or the W.K.B.J. solutions constitute a powerful technique for dealing with these approximations. Bluntly speaking, the technique is to restrict oneself to the first term of the asymptotic series; the extraordinary difficulties that arise by using such a simple approximation must be investigated and smoothed out by means of a series of rules by which such approximations may be manipulated without fear of mathematical error.

The use of approximations in mathematical physics is so familiar that it is surprising that some have complained about the point of view adopted in phase-integral methods. For example, with no just foundation for such remarks, Smyth [102] has criticized a paper making use of the method by writing: ‘It should be observed that the authors have used a solution which is a very poor approximation to the given problem as an approximate solution to another problem. It is certainly not to be expected that the results obtained in this manner will have any connection with the original problem.’ Concerning approximations, Schelkunoff [97] has more wisely remarked that there is ‘something in human nature that makes one yearn for the exact answer to a given problem. In particular it makes little difference whether a given problem is solved approximately or replaced by an approximating problem which is then solved exactly.’ Introducing a new approach that leads to a difference in applicability, Hines [59] has observed that his new method yields ‘an approximate evaluation of the exact solution, rather than an exact evaluation of an approximate solution as is found in the W.K.B.J. method’.

Phase-integral methods are applied to the particular equation

in which h is a large parameter and q(x, h) a variable function of x and h. The real independent variable x may often be replaced by the complex variable z. Under certain circumstances it will be shown that approximate solutions of this equation are

the error term of the form shown being maintained uniformly throughout certain restricted domains of the complex z-plane. A zero of q in the complex plane is called a transition point; evidently the expressions (1.2) cease to be valid solutions near a transition point owing to the singularity caused by the factor q−1/4, while the exact solution of equation (1.1) must obviously remain finite at such a point. These expressions are known as the phase-integral solutions or the W.K.B.J. solutions, after Wentzel [115] (1926), Kramers [72] (1926), Brillouin [21] (1926) and Jeffreys [64] (1923). Concerning the common name given to these solutions, Bailey [6] has chosen to call them ‘the L.R. approximations’ after Liouville [79] (1837) and Rayleigh [94] (1912). Moreover, Bailey has written: ‘The custom, based on historical ignorance, of using the titles B.K.W. or W.K.B. (or some other permutation of these three letters) is wrong as it does such flagrant injustice to the truth.’ Jeffreys [66] has often called these solutions ‘approximations of Green's type’, after Green [48] (1837).

The object of phase-integral methods is to show how such solutions may be traced about the complex z-plane beyond their original restricted domains of validity.

In connection with solution (1.2), it should be stressed that we may discuss errors between an approximate solution and an exact solution even when there exists no exact analytical solution of the differential equation. An exact solution exists, not because some analytical form can necessarily be found for it, but because what are known to mathematicians as existence theorems demonstrate the existence of such a solution; see, for example, Chapter XII of the text by Ince [127], Ordinary Differential Equations. The error term O(1/ h) occurring in solution (1.2) implies that an exact solution would contain a term F(z, h)/ h, where |F(z, h)| is bounded by the inequality |F(z, h)| < M, for all z in the domain under consideration and with h greater than some value h0. There is no implication that h must tend to infinity in (1.2); all we assert is that h must be large. On the other hand, z may or may not tend to infinity in the domain under consideration. Hence problems that require finite values of h and z are embraced by solution (1.2).

Historically, the use of such approximate solutions may be traced to Carlini (1817), whose work may be referred to in Watson [114] (page 6). Carlini considered a specific equation, being in effect Bessel's equation, for which he derived an approximate solution valid when n is large and for 0 < x < n. Liouville [79] (1837) and Green [48] (1837) produced asymptotic solutions for more general equations, but void of rigour, for a range of x in which no transition point occurs. Green was concerned with the propagation of long waves in a channel of non-uniform section, provided that the period is short enough for the depth and width of the channel not to vary greatly within a wavelength; he was thereby able to show that for tidal waves in such a channel energy is transmitted without reflection losses.

The name of Horn [60] (1899) should also be mentioned in this connection, though strictly speaking his name belongs to the particular investigators detailed in section 1.5. He is usually stated to be the original principal contributor to the theory of the existence of asymptotic solutions in intervals free from transition points. From the W.K.B.J. point of view, we may observe that Jeffreys [64] has remarked that Horn did not give the most convenient form for the first term of the solution.

1.2 The Stokes phenomenon

Sir George Gabriel Stokes (1819-1903) would have been further honoured had it been realized then that without the existence of the discontinuous changes in the arbitrary constants that occur in the asymptotic solutions of certain differential equations, the reflection of waves by inhomogeneous media (such as the ionosphere) would have been an impossibility. Stokes had been confronted by this phenomenon in his study of Bessel functions, and evidently was troubled with the difficulty for some years before finally getting to the root of the matter. On March 19th, 1857, Stokes [103] wrote to his young lady: ‘I have been doing what I guess you won't let me do when we are married, sitting up till 3 o'clock in the morning fighting hard against a mathematical difficulty. Some years ago I attacked an integral of Airy's, and after a severe trial reduced it to a readily calculable form. But there was one difficulty about it which, though I tried till I almost made myself ill, I could not get over, and at last I had to give it up and profess myself unable to master it. I took it up again a few days ago, and after two or three days' fight, the last of which I sat up till 3, I at last mastered it.’

In his first paper on the subject, Stokes [104] (1857) considered the equation

w˝ − 9 zw = 0

in the complex z-plane. (Nowadays, the factor 9 would be omitted.) He gave two independent power-series solutions of this equation, which for small |z| could be used in the computation of the general solution, which would involve two fixed arbitrary constants multiplying the two power series respectively. Secondly, he gave the two independent asymptotic expansions of the equation (the leading terms being proportional to our expression (1.2) with q = − 9 z and h = 1). He noticed that if for a certain range of arg z a general solution was represented by a certain linear combination of the two asymptotic solutions,