Stationary phase, method of the

A method for calculating the asymptotics of integrals of rapidly-oscillating functions:

$$ \tag{* } F( \lambda ) = \int\limits _ \Omega f( x) e ^ {i \lambda S( x) } dx, $$

where $ x \in \mathbf R ^ {n} $, $ \lambda > 0 $, $ \lambda \rightarrow + \infty $, is a large parameter, $ \Omega $ is a bounded domain, the function $ S( x) $ (the phase) is real, the function $ f( x) $ is complex, and $ f, S \in C ^ \infty ( \mathbf R ^ {n} ) $. If $ f \in C _ {0} ^ \infty ( \mathbf R ^ {n} ) $, i.e. $ f $ has compact support, and the phase $ S( x) $ does not have stationary points (i.e. points at which $ S ^ \prime ( x) = 0 $) on $ \supp f $, $ \Omega = \mathbf R ^ {n} $, then $ F( \lambda ) = O( \lambda ^ {- n } ) $, for all $ n $ as $ \lambda \rightarrow + \infty $. Therefore, when $ \lambda \rightarrow + \infty $, the points of stationary phase and the boundary $ \partial \Omega $ give the essential contribution to the asymptotics of the integral (*). The integrals

$$ V _ {x ^ {0} } ( \lambda ) = \ \int\limits _ \Omega f( x) \phi _ {0} ( x) e ^ {i \lambda S( x) } dx , $$

$$ V _ {\partial \Omega } ( \lambda ) = \int\limits _ \Omega f( x) \phi _ {\partial \Omega } ( x) e ^ {i \lambda S( x) } dx $$

are called the contributions from the isolated stationary point $ x ^ {0} $ and the boundary, respectively, where $ \phi _ {0} \in C _ {0} ^ \infty ( \Omega ) $, $ \phi _ {0} \equiv 1 $ near the point $ x ^ {0} $ and $ \supp \phi _ {0} $ does not contain any other stationary points, $ \phi _ {\partial \Omega } \in C _ {0} ^ \infty ( \mathbf R ^ {n} ) $ and $ \phi _ {\partial \Omega } \equiv 1 $ in a certain neighbourhood of the boundary. For $ n= 1 $, $ \Omega = ( a, b) $:

1) $ V _ {a} ( \lambda ) = \frac{i}{\lambda S ^ \prime ( a) } e ^ {i \lambda S( a) } [ f( a) + O( \lambda ^ {-1} )] $, if $ S ^ \prime ( a) \neq 0 $;

2)

$$ V _ {x ^ {0} } ( \lambda ) = \sqrt { \frac{2 \pi }{\lambda | S ^ {\prime\prime} ( x ^ {0} ) | } } e ^ {i ( \lambda S ( x ^ {0} ) + \pi \delta _ {0} / 4 ) } \times $$

$$ \times [ f( x ^ {0} ) + O( \lambda ^ {-1} )],\ \ \delta _ {0} = \mathop{\rm sgn} S ^ {\prime\prime} ( x ^ {0} ), $$

if $ x ^ {0} $ is an interior point of $ \Omega $ and $ S ^ \prime ( x ^ {0} ) = 0 $, $ S ^ {\prime\prime} ( x ^ {0} ) \neq 0 $.

Detailed research has been carried out in the case where $ n= 1 $, the phase $ S $ has a finite number of stationary points, all of finite multiplicity, and the function $ f $ has zeros of finite multiplicity at these points and at the end-points of an interval $ \Omega $. Asymptotic expansions have been obtained. The case where the functions $ f $ and $ S $ have power singularities has also been studied: for example, $ f = x ^ \alpha f _ {1} ( x) $, $ S = x ^ \beta S _ {1} ( x) $, where $ f _ {1} $, $ S _ {1} $ are smooth functions when $ x = 0 $, $ \alpha > - 1 $, $ \beta > 0 $.

Let $ n \geq 2 $, and let $ x ^ {0} \in \Omega $ be a non-degenerate stationary point (i.e. $ \Delta _ {S} ( x ^ {0} ) = \mathop{\rm det} S ^ {\prime\prime} ( x ^ {0} ) \neq 0 $). The contribution from the point $ x ^ {0} $ is then equal to

$$ V _ {x ^ {0} } ( \lambda ) = \ \left ( \frac{2 \pi } \lambda \right ) ^ {n/2} | \Delta _ {S} ( x ^ {0} ) | ^ {-1/2} \times $$

$$ \times \mathop{\rm exp} \left [ i \left ( \lambda S( x ^ {0} ) + + \frac \pi {4} \delta _ {S} ( x ^ {0} ) \right ) \right ] [ f( x ^ {0} ) + O( \lambda ^ {-1} )], $$

where $ \delta _ {S} ( x ^ {0} ) $ is the signature of the matrix $ S ^ {\prime\prime} ( x ^ {0} ) $. There is also an asymptotic series for $ V _ {x ^ {0} } ( \lambda ) $ (for the formulas of the contribution $ V _ {\partial \Omega } ( \lambda ) $ in the case of a smooth boundary, see [5]).

If $ x ^ {0} \in \Omega $ is a stationary point of finite multiplicity, then (see [6])

$$ V _ {x ^ {0} } ( \lambda ) \sim \mathop{\rm exp} [ i \lambda S( x ^ {0} )] \sum _ { k= 0} ^ \infty \left ( \sum _ { l= 0} ^ { N } a _ {kl} \lambda ^ {- r _ {k} } ( \mathop{\rm ln} \lambda ) ^ {l} \right ) , $$

where $ r _ {k} $ are rational numbers, $ n/2 \leq r _ {0} < \dots < r _ {k} \rightarrow + \infty $. Degenerate stationary points have been studied, cf. [3], [4].

Studies have been made on the case where the phase $ S = S( x, \alpha ) $ depends on a real parameter $ \alpha $, and for small $ | \alpha | $ has two close non-degenerate stationary points. In this case, the asymptotics of the integral $ F( \lambda , \alpha ) $ can be expressed in terms of Airy functions (see [5], [10]). The method of the stationary phase has an operator variant: $ \lambda = A $, where $ A $ is the infinitesimal operator of the strongly-continuous group $ \{ e ^ {itA} \} $ of operators bounded on the axis $ - \infty < t < \infty $, acting on a Banach space $ B $, and $ f( x) $, $ S( x) $ are smooth functions with values in $ B $[9]. If the functions are analytic, then the method of the stationary phase is a particular case of the saddle point method.

References

[1]	W. Thomson, Philos. Mag. , 23 (1887) pp. 252–255
[2]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956) MR0081379 MR0078494 Zbl 0072.11703 Zbl 0070.29002
[3]	E.Ya. Rieksteyn'sh, "Asymptotic expansions of integrals" , 1–2 , Riga (1974–1977) (In Russian)
[4]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974) MR0435697 Zbl 0308.41023 Zbl 0303.41035
[5]	M.V. Fedoryuk, "The method of steepest descent" , Moscow (1977) (In Russian)
[6]	M.F. Atiyah, "Resolution of singularities and division of distributions" Comm. Pure Appl. Math. , 23 : 2 (1970) pp. 145–150 MR0256156 Zbl 0188.19405
[7]	V.I. Arnol'd, "Remarks on the stationary phase method and Coxeter numbers" Russian Math. Surveys , 28 : 5 (1973) pp. 19–48 Uspekhi Mat. Nauk , 28 : 5 (1973) pp. 17–44 Zbl 0291.40005
[8]	A.N. Varchenko, "Newton polyhedra and estimation of oscillating integrals" Funct. Anal. Appl , 10 : 3 (1976) pp. 175–196 Funktsional. Anal. i Prilozhen. , 10 : 3 (1976) pp. 13–38 Zbl 0351.32011
[9]	V.P. Maslov, M.V. Fedoryuk, "Semi-classical approximation in quantum mechanics" , Reidel (1981) (Translated from Russian) Zbl 0458.58001
[10]	M.V. Fedoryuk, "Asymptotics. Integrals and series" , Moscow (1987) (In Russian) MR0950167 Zbl 0641.41001

Comments

An integral of the form (*) is a special case of a so-called oscillatory integral, or Fourier integral operator, cf. also [a2].

References

[a1]	R. Wong, "Asymptotic approximations of integrals" , Acad. Press (1989) MR1016818 Zbl 0679.41001
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. §7.7 MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001