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