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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

$$\int_Cf(z)e^{\lambda g(z)}dz$$

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was first published by Debye (1909), who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note Riemann (1863) about hypergeometric functions.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 Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the Riemann-Siegel formula.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 : Cn → C and C ⊂ Cn. If

$$M = \sup_{x \in C} \Re (S(x)) < \infty,$$

where \Re (\cdot) denotes the real part, and there exists a positive real number λ0 such that

$$\int_{C} \left| f(x) e^{\lambda_0 S(x)} \right| dx < \infty,$$

then the following estimate holds:

$$\left| \int_{C} f(x) e^{\lambda S(x)} dx \right| \leqslant \text{const}\cdot e^{\lambda M}, \qquad \forall \lambda \in \mathbb{R}, \quad \lambda \geqslant \lambda_0.$$

Proof of the simple estimate

\begin{align} \left| \int_{C} f(x) e^{\lambda S(x)} dx \right| &\leqslant \int_C |f(x)| \left|e^{\lambda S(x)} \right| dx \\ &\equiv \int_{C} |f(x)| e^{\lambda M} \left | e^{\lambda_0 (S(x)-M)} e^{(\lambda-\lambda_0)(S(x)-M)} \right| dx \\ &\leqslant \int_C |f(x)| e^{\lambda M} \left| e^{\lambda_0 (S(x)-M)} \right| dx && \left| e^{(\lambda-\lambda_0)(S(x) - M)} \right| \leqslant 1 \\ &= \underbrace{e^{-\lambda_0 M} \int_{C} \left| f(x) e^{\lambda_0 S(x)} \right| dx}_{\text{const}} \cdot e^{\lambda M}. \end{align}

The case of a single non-degenerate saddle point
Basic notions and notation

Let x be a complex n-dimensional vector, and

$$S''_{xx}(x) \equiv \left( \frac{\partial^2 S(x)}{\partial x_i \partial x_j} \right), \qquad 1\leqslant i,\, j\leqslant n,$$

denote the Hessian matrix for a function S(x). If

$$\boldsymbol{\varphi}(x) = (\varphi_1(x), \varphi_2(x), \ldots, \varphi_k(x))$$

is a vector function, then its Jacobian matrix is defined as

$$\boldsymbol{\varphi}_x' (x) \equiv \left( \frac{\partial \varphi_i (x)}{\partial x_j} \right), \qquad 1 \leqslant i \leqslant k, \quad 1 \leqslant j \leqslant n.$$

A non-degenerate saddle point, z0 ∈ Cn, of a holomorphic function S(z) is a point where the function reaches an extremum (i.e., ∇S(z0) = 0) and has a non-vanishing determinant of the Hessian (i.e., $$\det S''_{zz}(z^0) \neq 0).$$

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 z0 of a holomorphic function S(z), there exist coordinates in terms of which S(z) − S(z0) is quadratic. 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 \det S''_{zz}(z^0) \neq 0. 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

$$\forall w \in V: \qquad S(\boldsymbol{\varphi}(w)) = S(z^0) + \frac{1}{2} \sum_{j=1}^n \mu_j w_j^2, \quad \det\boldsymbol{\varphi}_w'(0) = 1,$$

Here, the μj are the eigenvalues of the matrix $$S_{zz}''(z^0).$$

An illustration of Complex Morse Lemma
Proof of Complex Morse Lemma

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;
$$\Re(S(z)) has a single maximum: \( \max_{z \in I_x} \Re(S(z)) = \Re(S(x^0))$$ for exactly one point x0 ∈ Ix;
x0 is a non-degenerate saddle point (i.e., ∇S(x0) = 0 and \det S''_{xx}(x^0) \neq 0).

Then, the following asymptotic holds

$$I(\lambda) \equiv \int_{I_x} f(x) e^{\lambda S(x)} dx = \left( \frac{2\pi}{\lambda}\right)^{\frac{n}{2}} e^{\lambda S(x^0)} \left(f(x^0)+ O\left(\lambda^{-1}\right) \right) \prod_{j=1}^n (-\mu_j)^{-\frac{1}{2}}, \qquad \lambda \to \infty,$$
(8)

where μj are eigenvalues of the Hessian S''_{xx}(x^0) and (-\mu_j)^{-\frac{1}{2}} are defined with arguments

$$\left | \arg\sqrt{-\mu_j} \right| < \tfrac{\pi}{4}.$$

(9)

This statement is a special case of more general results presented in Fedoryuk (1987).[4]
Derivation of equation (8)

Equation (8) can also be written as

$$I(\lambda) = \left( \frac{2\pi}{\lambda}\right)^{\frac{n}{2}} e^{\lambda S(x^0)} \left ( \det (-S_{xx}''(x^0)) \right )^{-\frac{1}{2}} \left (f(x^0) + O\left(\lambda^{-1}\right) \right),$$
(13)

where the branch of

$$\sqrt{\det \left (-S_{xx}''(x^0) \right)}$$

is selected as follows

\begin{align} \left (\det \left (-S_{xx}''(x^0) \right ) \right)^{-\frac{1}{2}} &= \exp\left( -i \text{ Ind} \left (- S_{xx}''(x^0) \right ) \right) \prod_{j=1}^n \left| \mu_j \right|^{-\frac{1}{2}}, \\ \text{Ind} \left (-S_{xx}''(x^0) \right) &= \tfrac{1}{2} \sum_{j=1}^n \arg (-\mu_j), && |\arg(-\mu_j)| < \tfrac{\pi}{2}. \end{align}

Consider important special cases:

If S(x) is real valued for real x and x0 in Rn (aka, the multidimensional Laplace method), then[7]

$$\text{Ind} \left(-S_{xx}''(x^0) \right ) = 0.$$

If S(x) is purely imaginary for real x (i.e., $$\Re(S(x)) = 0$$ for all x in Rn) and x0 in Rn (aka, the multidimensional stationary phase method),[8] then[9]

$$\text{Ind} \left (-S_{xx}''(x^0) \right ) = \frac{\pi}{4} \text{sign }S_{xx}''(x_0),$$

where $$\text{sign }S_{xx}''(x_0)$$ denotes the signature of matrix $$S_{xx}''(x_0),$$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.,

$$\nabla S \left (x^{(k)} \right ) = 0, \quad \det S''_{xx} \left (x^{(k)} \right ) \neq 0, \quad x^{(k)} \in \Omega_x^{(k)},$$

where

$$\left \{ \Omega_x^{(k)} \right \}_{k=1}^K$$

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

\begin{align} \sum_{k=1}^K \rho_k(x) &= 1, && \forall x \in \Omega_x, \\ \rho_k(x) &= 0 && \forall x \in \Omega_x\setminus \Omega_x^{(k)}. \end{align} Whence, \( \int_{I_x \subset \Omega_x} f(x) e^{\lambda S(x)} dx \equiv \sum_{k=1}^K \int_{I_x \subset \Omega_x} \rho_k(x) f(x) e^{\lambda S(x)} dx.

Therefore as λ → ∞ we have:

$$\sum_{k=1}^K \int_{\text{a neighborhood of }x^{(k)}} f(x) e^{\lambda S(x)} dx = \left(\frac{2\pi}{\lambda}\right)^{\frac{n}{2}} \sum_{k=1}^K e^{\lambda S \left (x^{(k)} \right )} \left ( \det \left(-S_{xx}'' \left (x^{(k)} \right )\right) \right)^{-\frac{1}{2}} f \left (x^{(k)} \right ),$$

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(z0) = 0 and $$\det S''_{zz}(z^0) = 0$$, the point z0 ∈ Cn is called a degenerate saddle point of a function S(z).

Calculating the asymptotic of

$$\int f(x) e^{\lambda S(x)} dx,$$

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.

Pearcey integral
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 I0(λ), given by equation (8), and a simple estimate for the discarded integral I1(λ).
This is justified by comparing the integral asymptotic over Rn [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 \left | \arg\sqrt{-\mu_j} \right | \leqslant \tfrac{\pi}{4}.

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): 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.

Some sources (e.g. Bell (2006)) consider H to be a densely embedded Hilbert subspace of the Banach space E, with i simply the inclusion of H into E. There is no loss of generality in taking this "embedded spaces" viewpoint instead of the "different spaces" viewpoint given above.

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