Method of steepest descent

A simple estimate

Let  f, S : Cⁿ → C and C ⊂ Cⁿ. If

M = sup x ∈ C ℜ ( S ( x ) ) < ∞ ,

where ℜ ( ⋅ ) denotes the real part, and there exists a positive real number λ₀ such that

∫ C | f ( x ) e λ 0 S ( x ) | d x < ∞ ,

then the following estimate holds:

| ∫ C f ( x ) e λ S ( x ) d x | ⩽ const ⋅ e λ M , ∀ λ ∈ R , λ ⩾ λ 0 .

Basic notions and notation

Let x be a complex n-dimensional vector, and

S x x ″ ( x ) ≡ ( ∂ 2 S ( x ) ∂ x i ∂ x j ) , 1 ⩽ i , j ⩽ n ,

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

φ ( x ) = ( φ 1 ( x ) , φ 2 ( x ) , … , φ k ( x ) )

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

φ x ′ ( x ) ≡ ( ∂ φ i ( x ) ∂ x j ) , 1 ⩽ i ⩽ k , 1 ⩽ j ⩽ 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 z z ″ ( z 0 ) ≠ 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 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 det S z z ″ ( z 0 ) ≠ 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

∀ w ∈ V : S ( φ ( w ) ) = S ( z 0 ) + 1 2 ∑ j = 1 n μ j w j 2 , det φ w ′ ( 0 ) = 1 ,

Here, the μ_j are the eigenvalues of the matrix S z z ″ ( z 0 ) .

The asymptotic expansion in the case of a single non-degenerate saddle point

Assume

1.  f (z) and S(z) are holomorphic functions in an open, bounded, and simply connected set Ω_x ⊂ Cⁿ such that the I_x = Ω_x ∩ Rⁿ is connected;
2. ℜ ( S ( z ) ) has a single maximum: max z ∈ I x ℜ ( S ( z ) ) = ℜ ( S ( x 0 ) ) for exactly one point x⁰ ∈ I_x;
3. x⁰ is a non-degenerate saddle point (i.e., ∇S(x⁰) = 0 and det S x x ″ ( x 0 ) ≠ 0 ).

Then, the following asymptotic holds

I ( λ ) ≡ ∫ I x f ( x ) e λ S ( x ) d x = ( 2 π λ ) n 2 e λ S ( x 0 ) ( f ( x 0 ) + O ( λ − 1 ) ) ∏ j = 1 n ( − μ j ) − 1 2 , λ → ∞ ,

where μ_j are eigenvalues of the Hessian S x x ″ ( x 0 ) and ( − μ j ) − 1 2 are defined with arguments

| arg ⁡ − μ j | < π 4 .

This statement is a special case of more general results presented in Fedoryuk (1987).

Equation (8) can also be written as

I ( λ ) = ( 2 π λ ) n 2 e λ S ( x 0 ) ( det ( − S x x ″ ( x 0 ) ) ) − 1 2 ( f ( x 0 ) + O ( λ − 1 ) ) ,

where the branch of

det ( − S x x ″ ( x 0 ) )

is selected as follows

( det ( − S x x ″ ( x 0 ) ) ) − 1 2 = exp ⁡ ( − i  Ind ( − S x x ″ ( x 0 ) ) ) ∏ j = 1 n | μ j | − 1 2 , Ind ( − S x x ″ ( x 0 ) ) = 1 2 ∑ j = 1 n arg ⁡ ( − μ j ) , | arg ⁡ ( − μ j ) | < π 2 .

Consider important special cases:

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

If S(x) is purely imaginary for real x (i.e., ℜ ( S ( x ) ) = 0 for all x in Rⁿ) and x⁰ in Rⁿ (aka, the multidimensional stationary phase method), then

where sign  S x x ″ ( x 0 ) denotes the signature of matrix S x x ″ ( 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.,

∇ S ( x ( k ) ) = 0 , det S x x ″ ( x ( k ) ) ≠ 0 , x ( k ) ∈ Ω x ( k ) ,

where

{ Ω x ( k ) } k = 1 K

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

∑ k = 1 K ρ k ( x ) = 1 , ∀ x ∈ Ω x , ρ k ( x ) = 0 ∀ x ∈ Ω x ∖ Ω x ( k ) .

Whence,

∫ I x ⊂ Ω x f ( x ) e λ S ( x ) d x ≡ ∑ k = 1 K ∫ I x ⊂ Ω x ρ k ( x ) f ( x ) e λ S ( x ) d x .

Therefore as λ → ∞ we have:

∑ k = 1 K ∫ a neighborhood of  x ( k ) f ( x ) e λ S ( x ) d x = ( 2 π λ ) n 2 ∑ k = 1 K e λ S ( x ( k ) ) ( det ( − S x x ″ ( x ( k ) ) ) ) − 1 2 f ( x ( k ) ) ,

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 z z ″ ( z 0 ) = 0 , the point z⁰ ∈ Cⁿ is called a degenerate saddle point of a function S(z).

Calculating the asymptotic of

∫ f ( x ) e λ S ( x ) d x ,

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. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.