Method of steepest descent
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 (saddle 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 (approximately) real and has a maximum at the zero.
The method of steepest descent was first published by Debye (1909), who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note Riemann (1863) about hypergeometric functions. The contour of steepest descent has a minimax property, see Fedoryuk (2001). Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the Riemann-Siegel formula.
A simple estimate^{[1]}
Let f, S : C^{n} → C and C ⊂ C^{n}. If
where denotes the real part, and there exists a positive real number λ_{0} 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, z^{0} ∈ C^{n}, of a holomorphic function S(z) is a point where the function reaches an extremum (i.e., ∇S(z^{0}) = 0) and has a non-vanishing determinant of the Hessian (i.e., ).
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^{[2]} for holomorphic functions: near a non-degenerate saddle point z^{0} of a holomorphic function S(z), there exist coordinates in terms of which S(z) − S(z^{0}) is exactly quadratic. To make this precise, let S be a holomorphic function with domain W ⊂ C^{n}, and let z^{0} in W be a non-degenerate saddle point of S, that is, ∇S(z^{0}) = 0 and . Then there exist neighborhoods U ⊂ W of z^{0} and V ⊂ C^{n} of w = 0, and a bijective holomorphic function φ : V → U with φ(0) = z^{0} such that
Here, the μ_{j} are the eigenvalues of the matrix .
The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in.^{[3]} We begin by demonstrating
- Auxiliary Statement. Let f : C^{n} → C be holomorphic in a neighborhood of the origin and f (0) = 0. Then in some neighborhood, there exist functions g_{i} : C^{n} → C such that
- where each g_{i} is holomorphic and
From the identity
we conclude that
and
Without loss of generality, we translate the origin to z^{0}, such that z^{0} = 0 and S(0) = 0. Using the Auxiliary Statement, we have
Since the origin is a saddle point,
we can also apply the Auxiliary Statement to the functions g_{i}(z) and obtain
- (1)
Recall that an arbitrary matrix A can be represented as a sum of symmetric A^{(s)} and anti-symmetric A^{(a)} matrices,
The contraction of any symmetric matrix B with an arbitrary matrix A is
i.e., the anti-symmetric component of A does not contribute because
Thus, h_{ij}(z) in equation (1) can be assumed to be symmetric with respect to the interchange of the indices i and j. Note that
hence, det(h_{ij}(0)) ≠ 0 because the origin is a non-degenerate saddle point.
Let us show by induction that there are local coordinates u = (u_{1}, ... u_{n}), z = ψ(y), 0 = ψ(0), such that
- (3)
First, assume that there exist local coordinates y = (y_{1}, ... y_{n}), z = φ(y), 0 = φ(0), such that
- (4)
where H_{ij} is symmetric due to equation (2). By a linear change of the variables (y_{r}, ... y_{n}), we can assure that H_{rr}(0) ≠ 0. From the chain rule, we have
Therefore:
whence,
The matrix (H_{ij}(0)) can be recast in the Jordan normal form: (H_{ij}(0)) = LJL^{−1}, were L gives the desired non-singular linear transformation and the diagonal of J contains non-zero eigenvalues of (H_{ij}(0)). If H_{ij}(0) ≠ 0 then, due to continuity of H_{ij}(y), it must be also non-vanishing in some neighborhood of the origin. Having introduced , we write
Motivated by the last expression, we introduce new coordinates z = η(x), 0 = η(0),
The change of the variables y ↔ x is locally invertible since the corresponding Jacobian is non-zero,
Therefore,
- (5)
Comparing equations (4) and (5), we conclude that equation (3) is verified. Denoting the eigenvalues of by μ_{j}, equation (3) can be rewritten as
- (6)
Therefore,
- (7)
From equation (6), it follows that . The Jordan normal form of reads , where J_{z} is an upper diagonal matrix containing the eigenvalues and det P ≠ 0; hence, . We obtain from equation (7)
If , then interchanging two variables assures that .
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 Ω_{x} ⊂ C^{n} such that the I_{x} = Ω_{x} ∩ R^{n} is connected;
- has a single maximum: for exactly one point x^{0} ∈ I_{x};
- x^{0} is a non-degenerate saddle point (i.e., ∇S(x^{0}) = 0 and ).
Then, the following asymptotic holds
- (8)
where μ_{j} are eigenvalues of the Hessian and are defined with arguments
This statement is a special case of more general results presented in Fedoryuk (1987).^{[4]}
First, we deform the contour I_{x} into a new contour passing through the saddle point x^{0} and sharing the boundary with I_{x}. This deformation does not change the value of the integral I(λ). We employ the Complex Morse Lemma to change the variables of integration. According to the lemma, the function φ(w) maps a neighborhood x^{0} ∈ U ⊂ Ω_{x} onto a neighborhood Ω_{w} containing the origin. The integral I(λ) can be split into two: I(λ) = I_{0}(λ) + I_{1}(λ), where I_{0}(λ) is the integral over , while I_{1}(λ) is over (i.e., the remaining part of the contour I′_{x}). Since the latter region does not contain the saddle point x^{0}, the value of I_{1}(λ) is exponentially smaller than I_{0}(λ) as λ → ∞;^{[5]} thus, I_{1}(λ) is ignored. Introducing the contour I_{w} such that , we have
- (10)
Recalling that x^{0} = φ(0) as well as , we expand the pre-exponential function into a Taylor series and keep just the leading zero-order term
- (11)
Here, we have substituted the integration region I_{w} by R^{n} because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term.^{[6]} The integrals in the r.h.s. of equation (11) can be expressed as
- (12)
From this representation, we conclude that condition (9) must be satisfied in order for the r.h.s. and l.h.s. of equation (12) to coincide. According to assumption 2, is a negatively defined quadratic form (viz., ) implying the existence of the integral , which is readily calculated
Equation (8) can also be written as
- (13)
where the branch of
is selected as follows
Consider important special cases:
- If S(x) is real valued for real x and x^{0} in R^{n} (aka, the multidimensional Laplace method), then^{[7]}
- If S(x) is purely imaginary for real x (i.e., for all x in R^{n}) and x^{0} in R^{n} (aka, the multidimensional stationary phase method),^{[8]} then^{[9]}
- 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 (as well as in optics), Ind is related to the Maslov index see, e.g., Chaichian & Demichev (2001) and Schulman (2005).
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 Ω_{x}, 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 ρ_{k}(x) : Ω_{x} → [0, 1], 1 ≤ k ≤ K, such that
Whence,
Therefore as λ → ∞ we have:
where equation (13) was utilized at the last stage, and the pre-exponential function f (x) at least must be continuous.
The other cases
When ∇S(z^{0}) = 0 and , the point z^{0} ∈ C^{n} 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 (1978) and Fedoryuk (1987).
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 (see, e.g., Fedoryuk (1987) and Wong (1989)).
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 (properly speaking) 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
Notes
- ↑ A modified version of Lemma 2.1.1 on page 56 in Fedoryuk (1987).
- ↑ Lemma 3.3.2 on page 113 in Fedoryuk (1987)
- ↑ Poston & Stewart (1978), page 54; see also the comment on page 479 in Wong (1989).
- ↑ Fedoryuk (1987), pages 417-420.
- ↑ This conclusion follows from a comparison between the final asymptotic for I_{0}(λ), given by equation (8), and a simple estimate for the discarded integral I_{1}(λ).
- ↑ This is justified by comparing the integral asymptotic over R^{n} [see equation (8)] with a simple estimate for the altered part.
- ↑ See equation (4.4.9) on page 125 in Fedoryuk (1987)
- ↑ Rigorously speaking, this case cannot be inferred from equation (8) because the second assumption, utilized in the derivation, is violated. To include the discussed case of a purely imaginary phase function, condition (9) should be replaced by
- ↑ See equation (2.2.6') on page 186 in Fedoryuk (1987)
References
- Chaichian, M.; Demichev, A. (2001), Path Integrals in Physics Volume 1: Stochastic Process and Quantum Mechanics, Taylor & Francis, p. 174, ISBN 075030801X
- Debye, P. (1909), "Näherungsformeln für die Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche Werte des Index", Mathematische Annalen, 67 (4): 535–558, doi:10.1007/BF01450097 English translation in Debye, Peter J. W. (1954), 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. (1993), "A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation", Ann. of Math., The Annals of Mathematics, Vol. 137, No. 2, 137 (2), pp. 295–368, doi:10.2307/2946540, JSTOR 2946540.
- Erdelyi, A. (1956), Asymptotic Expansions, Dover.
- Fedoryuk, M V (2001), "Saddle_point_method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
- Fedoryuk, M. V. (1987), Asymptotic: Integrals and Series, Nauka, Moscow [in Russian].
- Kamvissis, S.; McLaughlin, K. T.-R.; Miller, P. (2003), "Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation", Annals of Mathematics Studies, Princeton University Press, 154.
- Riemann, B. (1863), Sullo svolgimento del quoziente di due serie ipergeometriche in frazione continua infinita (Unpublished note, reproduced in Riemann's collected papers.)
- Siegel, C. L. (1932), "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. (1978), Catastrophe Theory and Its Applications, Pitman.
- Schulman, L. S. (2005), "Ch. 17: The Phase of the Semiclassical Amplitude", Techniques and Applications of Path Integration, Dover, ISBN 0486445283
- Wong, R. (1989), Asymptotic approximations of integrals, Academic Press.