In mathematics, the method of steepest descent or stationary phase method or
saddle-point method is an extension of Laplace's method for approximating an
integral, where one deforms a contour integral in the complex plane to pass
near a stationary point, in roughly the direction of steepest descent or
stationary phase. The saddle-point approximation is used with integrals in
the complex plane, whereas Laplace’s method is used with real integrals.
The integral to be estimated is often of the form
where C is a contour and λ is large. One version of the method of steepest
descent deforms the contour of integration so that it passes through a
zero of the derivative g′(z) in such a way that on the contour g is real and
has a maximum at the zero. The method of steepest descent was first
published by Debye, who used it to estimate Bessel functions and pointed
out that it occurred in the unpublished note Riemann about hypergeometric
functions. The contour of steepest descent has a minimax property, see
Fedoryuk. Siegel described some other unpublished notes of Riemann, where he
used this method to derive the Riemann-Siegel formula.
A simple estimate Let f, S : Cn → C and C ⊂ Cn. If
where denotes the real part, and there exists a positive real number λ0 such
that then the following estimate holds:
The case of a single non-degenerate saddle point
= Basic notions and notation= Let x be a complex n-dimensional vector,
and denote the Hessian matrix for a function
S(x). If is a vector function, then its Jacobian
matrix is defined as A non-degenerate saddle point, z0 ∈ Cn,
of a holomorphic function S(z) is a point where the function reaches an
extremum = 0) and has a non-vanishing determinant of the Hessian.
The following is the main tool for constructing the asymptotics of
integrals in the case of a non-degenerate saddle point:
= Complex Morse Lemma= The Morse lemma for real-valued
functions generalizes as follows for holomorphic functions: near a
non-degenerate saddle point z0 of a holomorphic function S(z), there exist
coordinates in terms of which S(z) − S(z0) is exactly quadratic. To make this
precise, let S be a holomorphic function with domain W ⊂ Cn, and let z0 in W be a
non-degenerate saddle point of S, that is, ∇S(z0) = 0 and . Then there exist
neighborhoods U ⊂ W of z0 and V ⊂ Cn of w = 0, and a bijective holomorphic
function φ : V → U with φ(0) = z0 such that
Here, the μj are the eigenvalues of the matrix .
= The asymptotic expansion in the case of a single non-degenerate saddle point=
Assume f (z) and S(z) are holomorphic
functions in an open, bounded, and simply connected set Ωx ⊂ Cn such that
the Ix = Ωx ∩ Rn is connected; has a single maximum: for exactly one
point x0 ∈ Ix; x0 is a non-degenerate saddle point = 0
and ). Then, the following asymptotic holds
where μj are eigenvalues of the Hessian and are defined with arguments
This statement is a special case of more general results presented in Fedoryuk.
Equation can also be written as where the branch of
is selected as follows Consider important special cases:
If S(x) is real valued for real x and x0 in Rn, then
If S(x) is purely imaginary for real x and x0 in Rn, then
where denotes the signature of matrix , which equals to the number of negative
eigenvalues minus the number of positive ones. It is noteworthy that in
applications of the stationary phase method to the multidimensional WKB
approximation in quantum mechanics, Ind is related to the Maslov index see,
e.g., Chaichian & Demichev and Schulman. The case of multiple non-degenerate
saddle points If the function S(x) has multiple
isolated non-degenerate saddle points, i.e.,
where is an open cover of Ωx, then the
calculation of the integral asymptotic is reduced to the case of a singe saddle
point by employing the partition of unity. The partition of unity allows us
to construct a set of continuous functions ρk(x) : Ωx → [0, 1], 1 ≤ k ≤
K, such that Whence,
Therefore as λ → ∞ we have: where equation was utilized at the last
stage, and the pre-exponential function f (x) at least must be continuous.
The other cases When ∇S(z0) = 0 and , the point z0 ∈ Cn
is called a degenerate saddle point of a function S(z).
Calculating the asymptotic of when λ → ∞, f (x) is continuous, and
S(z) has a degenerate saddle point, is a very rich problem, whose solution
heavily relies on the catastrophe theory. Here, the catastrophe theory
replaces the Morse lemma, valid only in the non-degenerate case, to transform
the function S(z) into one of the multitude of canonical representations.
For further details see, e.g., Poston & Stewart and Fedoryuk.
Integrals with degenerate saddle points naturally appear in many applications
including optical caustics and the multidimensional WKB approximation in
quantum mechanics. The other cases such as, e.g., f (x)
and/or S(x) are discontinuous or when an extremum of S(x) lies at the integration
region's boundary, require special care and Wong).
Extensions and generalizations An extension of the steepest descent
method is the so-called nonlinear stationary phase/steepest descent
method. Here, instead of integrals, one needs to evaluate asymptotically
solutions of Riemann–Hilbert factorization problems.
Given a contour C in the complex sphere, a function f defined on that contour and
a special point, say infinity, one seeks a function M holomorphic away from the
contour C, with prescribed jump across C, and with a given normalization at
infinity. If f and hence M are matrices rather than scalars this is a problem
that in general does not admit an explicit solution.
An asymptotic evaluation is then possible along the lines of the linear
stationary phase/steepest descent method. The idea is to reduce
asymptotically the solution of the given Riemann–Hilbert problem to that of a
simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's
theorem is used to justify deformations of the jump contour.
The nonlinear stationary phase was introduced by Deift and Zhou in 1993,
based on earlier work of the Russian mathematician Alexander Its. A nonlinear
steepest descent method was introduced by Kamvissis, K. McLaughlin and P.
Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and
Zhou. As in the linear case, steepest descent contours solve a min-max
problem. The nonlinear stationary phase/steepest
descent method has applications to the theory of soliton equations and
integrable models, random matrices and combinatorics.
See also Pearcey integral
Notes References
Chaichian, M.; Demichev, A., Path Integrals in Physics Volume 1:
Stochastic Process and Quantum Mechanics, Taylor & Francis, p. 174,
ISBN 075030801X Debye, P., "Näherungsformeln für die
Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche
Werte des Index", Mathematische Annalen 67: 535–558, doi:10.1007/BF01450097
English translation in Debye, Peter J. W., The collected papers of Peter J. W.
Debye, Interscience Publishers, Inc., New York, ISBN 978-0-918024-58-9, MR
0063975 Deift, P.; Zhou, X., "A steepest descent
method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV
equation", Ann. Of Math. 137, pp. 295–368, doi:10.2307/2946540, JSTOR
2946540 . Erdelyi, A., Asymptotic Expansions,
Dover . Fedoryuk, M V, "Saddle_point_method", in
Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4 . Fedoryuk, M. V., Asymptotic: Integrals
and Series, Nauka, Moscow [in Russian]. Kamvissis, S.; McLaughlin, K. T.-R.;
Miller, P., "Semiclassical Soliton Ensembles for the Focusing Nonlinear
Schrödinger Equation", Annals of Mathematics Studies 154 .
Riemann, B., Sullo svolgimento del quoziente di due serie ipergeometriche
in frazione continua infinita Siegel, C. L., "Über Riemanns Nachlaß
zur analytischen Zahlentheorie", Quellen Studien zur Geschichte der Math. Astron.
und Phys. Abt. B: Studien 2: 45–80 Reprinted in Gesammelte Abhandlungen,
Vol. 1. Berlin: Springer-Verlag, 1966. Poston, T.; Stewart, I., Catastrophe
Theory and Its Applications, Pitman . Schulman, L. S., "Ch. 17: The Phase of
the Semiclassical Amplitude", Techniques and Applications of Path Integration,
Dover, ISBN 0486445283 Wong, R., Asymptotic approximations of
integrals, Academic Press .