Retracing our steps we see that for z in region mathrm{II}, say, of 211.x,

載入超時，點擊重試

(219.1)

Y(z) = U(z) e^{nsigma_3 g(z)}

= e^{frac{n}{2} l sigma_3} W(z) e^{-frac{n}{2} l sigma_3} e^{n sigma_3 g(z)}

= e^{frac{n}{2} l sigma_3} hat{W}(z) egin{pmatrix} 1 & 0 \ e^{-nS(z)} & 1 end{pmatrix} e^{-frac{n}{2} l sigma_3} e^{n sigma_3 g(z)}

underset{n ightarrow infty}{sim} e^{frac{n}{2} l sigma_3} egin{pmatrix} frac{eta + eta^{-1}}{2} & frac{eta - eta^{-1}}{2i} \ frac{eta^{-1} - eta}{2i} & frac{eta + eta^{-1}}{2} end{pmatrix} egin{pmatrix} 1 & 0 \ e^{-nS} & 1 end{pmatrix} e^{-frac{n}{2} l sigma_3} e^{n sigma_3 g(z)}

and we obtain, in particular, the asymptotics of pi_n(z) = Y_{11}(z) as n ightarrow infty. There are, of course, analogous formulae for z in the other 3 regions.

Remark 219+ ightarrow

A RHP (Sigma, v) can be converted into a problem of singular integral equations on Sigma in the following way.

Remark 219+

We see here a fundamentally new phenomena for the non-commutative steepest-descent method that is not present in the classical case: In the classical case, the leading order contribution to the integral comes from the stationary phase point, or a finite # of stationary phase points. But in the non-commutative case, we see that the leading order contribution to the RHP comes from a whole continuum of points, viz, Sigma = (-a, a).

Recall v in L^infty. Let

C_v f = int_Sigma frac{f(s) (v(s) - I)}{s - z} frac{ds}{2 pi i}

i.e. C_v f = C_Sigma (f(v - I)) where C_Sigma is the Cauchy operator for Sigma. Then for the contours, we are considering (a finite union of smooth arcs, with a finite # of points of self-intersection), and certainly for far more general contours (Carleson curves!), we have for f in L^2(Sigma)

C_{v_pm} f(z) = lim_{z^prime ightarrow z_pm} (C_v f)(z^prime) = lim_{z^prime ightarrow z_pm} C_Sigma (f(v - I)(z)) = C_{Sigma_pm} (f(v - I))

exists almost everywhere as non-tangential limits.

Moreover, ||C_{v_pm} f||_{L^2(Sigma)} le c ||f||_{L^2}.

Also,

C_{Sigma_+} f(z) - C_{Sigma_-} f(z) = f(z), almost everywhere z in Sigma.

(Supplemental information on RHPs in the sense of L^p(Sigma), 1 < p < infty)

Remark 220+:

In order to make the asymptotic calculations for the RHP valid, we must define what it means to solve a RHP in the sense of L^p(Sigma), 1 < p < infty.

(see, for example, P. Deift and X. Zhou CPAM 56, 2003, #8, 1029-1077, or ArXiv 0206224)

Let 1 < p < infty.

We say a pair of (matrix-valued) function m_pm in L^p(Sigma) belong to partial C_p(Sigma) if m_pm = C_Sigma^pm h for some h in L^p(Sigma).

Here C_Sigma^pm are the boundary values of the Cauchy transform on Sigma as before. Note that

h = C_Sigma^+ h - C_Sigma^- h = m_+ - m_-

so h is uniquely determined by m_pm.

We say that m_pm solves the normalized RHP (Sigma, v) in the sense of if m_pm - I in partial C_p(Sigma),

(220+1.1)

m_pm = I + C^pm h

for some h in L^p(Sigma), and

(220+1.2)

m_+ = m_- v almost everywhere on Sigma.

We say that

m(z) = I + C_Sigma h(z) = I + frac{1}{2 pi i} int_Sigma frac{h(s)}{s - z} ds

is the (analytic) extension of m_pm off Sigma. Note that

  • m(z) is analytic in mathbb{C} ackslash Sigma;
  • m_+ = m_- v almost everywhere on Sigma;
  • m(z) ightarrow I (in some sense) as z ightarrow infty.

so that m_pm above produces a solution of the RHP in the "natural" sense.

In order to solve the RHP (Sigma, v) with v, v^{-1} in GL(n, mathbb{C}) and v - I in L^p(Sigma), we proceed as follows: Let mu in I + L^p(Sigma) solve the equation

(220+2.1)

(1 - C_v) mu = I = I_n

on Sigma.

Here C_v h = C^- (h(v - I)) and I is the function identically = I on Sigma. Clearly ||C_v h||_p le ||C^-||_p ||v - I||_infty ||h||_p.

We always assume Sigma is Carleson so that C^pm are bounded in L^p(Sigma), ||C^pm||_p le infty. Hence C_v in mathfrak{h}(L^p).

More precisely, if we write mu = 1 + lambda, lambda in L^p, then (220+2.1) takes the form

(220+3.1)

(1 - C_v) lambda = C_v I = C^- (v - I) in L^p

which is an equation in L^p. To solve (220+3.1), and hence (220+2.1), we need to know that

(220+3.2)

(1 - C_v)^{-1} exists in L^p(Sigma).

Under this assumption, set

(220+3.3)

m_pm = I + C^pm mu(v - I).

Now, using (220+2.1),

(220+3.4(i))

m_+ = I + C^+ mu(v - I)

= I + C^- mu(v - I) + mu(v - I)

= I + C_v mu + mu(v - I)

= mu + mu(v - I) = mu v

and

(220+3.4(ii))

m_- = I + C^- mu(v - I) = I + C_v mu = mu

and hence

(220+3.5)

m_+ = m_- v.

But as

mu(v - I) = (v - I) + lambda(v - I) in L^p,

we see from (220+3.3) that m_pm in I + partial C_p and together with (220+3.4), we see that m_pm solve the normalized RHP (Sigma, v) in the sense of L^p.

Thus the analysis of the RHP boils down to the analysis of the (singular integral) equation (220+2.1).

Now suppose that we have a sequence (Sigma, v_n) of normalized RHPs on a fixed contour Sigma, and suppose that as n ightarrow infty, v_n ightarrow v, at least pointwise almost everywhere on Sigma. How much more do we need to know about the convergence to ensure that the solutions m_{n pm} of the RHPs (Sigma, v_n) converge to the solution m_pm of the (normalized) RHP (Sigma, v)?

From (220+3.4(i)) (220+3.4(ii)), we have

m_{n+} = mu_n v_n, m_{n-} = mu_n;

m_+ = mu v, m_- = mu.

So to obtain convergence in L^p(Sigma), we would certainly need to know that

(220+5.1)

mu_n - mu ightarrow 0 in L^p.

As m_{n+} - m_+ = mu_n v_n - mu v = (mu_n - mu)v - (v_n - v)mu, we have (v_n - v)mu = (v_n - v) + (v_n - v) lambda, lambda in L^p, so ||(v_n - v) mu||_{L^p} ightarrow 0 if

(a) v_n ightarrow v in L^p and by Dominated Convergence;

(b) sup ||v_n|| < infty.

Now from (220+3.1),

mu_n - mu = lambda_n - lambda = frac{1}{1 - C_{v_n}} C^- (v_n - I) - frac{1}{1 - C_v} C^-(v - I)

= frac{1}{1 - C_{v_n}} C^-(v_n -v) + frac{1}{1 - C_{v_n}} C^-(v_n - I) - frac{1}{1 - C_v} C^-(v - I)

= frac{1}{1 - C_{v_n}} C^-(v_n - v) + (frac{1}{1 - C_{v_n}} - frac{1}{1 - C_v}) C^-(v - I)

= frac{1}{1 - C_{v_n}} C^-(v_n - v) + (frac{1}{1 - C_{v_n}} C_{v_n - v} frac{1}{1 - C_v}) C^-(v - I).

Together with the above assumption (a), we have

frac{1}{1 - C_{v_n}} C^-(v_n - v) ightarrow 0 in L^p(Sigma)

if

(c) ||frac{1}{1 - C_{v_n}}|| le k < infty, forall n = 1, 2, dots.

In order for the second term to go to zero

(frac{1}{1 - C_{v_n}} C_{v_n - v} frac{1}{1 - C_v}) C^-(v - I) ightarrow 0 in L^p(Sigma),

we need, in addition to (c),

||C_{v_n - v}||_{L^p} ightarrow 0.

But ||C_{v_n - v} h|| = ||C^- h(v_n - v)|| le ||C^-||_p ||v_n - v||_infty ||h||_p

and so C_{v_n - v} ightarrow 0 if

(d) ||v_n - v||_infty ightarrow 0.

Standard manipulations show that if

||v_n - v||_infty ightarrow 0

and (1 - C_v)^{-1} exists, then (1 - C_{v_n})^{-1} for n sufficiently large. Assembling the above calculations we see that in order for m_{npm} ightarrow m_pm in L^p, we need

(220+7.1)

||v_n - v||_{L^p(Sigma)} + ||v_n - v||_{L^infty} ightarrow 0

and

(1 - C_v)^{-1} exists in L^p(Sigma).

This is why we cannot conclude that hat{W} ightarrow W_infty in Lecture 13! We do not have ||hat{v} - v_infty||_infty ightarrow 0 as n ightarrow infty.

Assume as before that v, v^{-1} in L^infty(Sigma) and in addition,

(221.0)

v - I in L^2(Sigma).

Suppose that

(221.1)

1 - C_{v-} is invertible in L^2(Sigma)

and let

(221.2)

mu = I + v

solve the equation

(221.3)

(1 - C_{v-}) mu = I

or more precisely,

(1 - C_{v-}) v = C_{v-} I = C_{Sigma^-} (v - I) in L^2.

Then

(221.4)

m(z) equiv I + int_Sigma frac{mu(s) (v(s) - I)}{s - z} frac{ds}{2 pi i}, z in mathbb{C} ackslash Sigma

solves the normalized RHP (Sigma, v). (Clearly m(z) is analytic in mathbb{C} ackslash Sigma.) Indeed, for z in Sigma,

m_+(z) = I + C_{v+}(mu) = I + mu(v - I) + C_{v-}(mu) = mu v.

Also m_-(z) = I + C_{v-} mu = mu.

Hence

m_+ = m_- v(z), z in Sigma.

Also, clearly, in some appropriate sense,

m(z) = I + mathcal{O}(frac{1}{z}) as z ightarrow infty,

by (221.4). Thus m(z) solves the normalized RHP (Sigma, v).

For z in Sigma, we see that

m_-(z) = mu(z) = frac{1}{1 - C_{v-}} I = I + frac{1}{1 - C_{v-}} (C_{Sigma-}(v - I)).

So we see now clearly that we would need hat{v} ightarrow v_infty in L^2 cap L^infty - to conclude that hat{W} ightarrow W_infty! We know hat{v} ightarrow v_infty in L^2, and point-wise almost everywhere on Sigma, but not in L^infty(Sigma)!

(For more details, see Remark 220+)

In order to show hat{W} ightarrow W_infty, we need more information about what is happening at the points z = pm a. We proceed as follows.

Extend the RHP (hat{Sigma}, hat{v}) for hat{W} on (212.x), to a RHP (hat{Sigma}_e, hat{v}_e) by adding in 2 small circles C_{pm a} around pm a

and set

(223.0)

hat{W}_e(z) = hat{W}(z), z in mathbb{C} ackslash hat{Sigma}_e

Thus

hat{v}_e (z) = hat{v}(z) for z in hat{Sigma} subset hat{Sigma}_e

and

hat{v}_e(z) = I for z in C_a cup C_{-a}.

We then construct explicitly a parametrix hat{W}_p(s) with the following properties:

(223.1)

hat{W}_p(z) = W_infty(z) for z outside the "dumbbell" region

(224.1)

hat{W}_p(z)_+ = hat{W}_p(z)_- hat{v}(z), z in (hat{Sigma} cap overset{circ}{C}_{+a}) cup (hat{Sigma} cap overset{circ}{C}_{-a})

where overset{circ}{C}_{pm a} denote the interior of C_{pm a} respectively:

(224.1)

hat{W}_{p+}(z) = hat{W}_{p-}(z) hat{v}_p(z), z in C_{pm a}

where hat{v}_p(z) = I + mathcal{O}_n(z), ||mathcal{O}_n||_{L^infty (C_{pm a})} = mathcal{O}(frac{1}{n}).

Note that on C_{pm a}, hat{W}_{p-}(z) = W_infty (z) and so (224.1) is a condition on the construction of hat{W}_p(z) for z in (overset{circ}{C}_{+a} cup overset{circ}{C}_{-a}) ackslash hat{Sigma}.

It follows, in particular from the above that hat{W}_p(z) solves a normalized RHP (hat{W}_e, v_p) where

Thus hat{W}_e and hat{W}_p have precisely the same jumps on longleftarrow longrightarrow. Hence R = hat{W}_e hat{W}_p^{-1} solves a normalized RHP with jumps on

On

R_+ = hat{W}_{e+} hat{W}_{p+}^{-1} = hat{W}_{e-} hat{v}^{-1} hat{W}_{p-}^{-1} = R_- hat{W}_{p-} hat{v}^{-1} hat{W}_{p-}^{-1} as hat{W}_p is analytic across these arcs. But as points on these arcs are at a finite distance from pm a, ||hat{v}(z) - I||_{L^infty (- overset{cap}{cup} -)} is exponentially decreasing, and hence the same is true foe v_R = hat{W}_{p-} hat{v}^{-1} hat{W}_{p-}^{-1} = W_infty hat{v}^{-1} W_infty^{-1}.

On the other hand, on

R_+(z) = hat{W}_{e+} hat{W}_{p+}^{-1} = hat{W}_{e-} hat{W}_{p-}^{-1} hat{W}_{p-} hat{v}_p^{-1} hat{W}_{p-}^{-1}= R_- v_R, v_R = hat{W}_{p-} hat{v}_p^{-1} hat{W}_{p-}^{-1} = W_infty v_p^{-1} W_infty^{-1} as hat{W}_e(z) is analytic across these arcs, and as ||hat{v}_p - I||_{L^infty (C_{pm a})} = mathcal{O}(frac{1}{n}), the same is true for v_R.

It follows then that

and so

R = hat{W} hat{W}_p^{-1} ightarrow I + mathcal{O}(frac{1}{n}) in L^2(hat{Sigma}_e)

i.e.

hat{W} = (I + mathcal{O}(frac{1}{n})) hat{W}_p.

As hat{W}_p is known explicitly from the construction, this provides the asymptotics of hat{W} as n ightarrow infty, and hence of Y(z), by (219.1).

We will show how the construct hat{W}_p(z) (using Airy functions!) in C_{pm a} ackslash hat{Sigma}, and make the above argument precise, in the next lecture.


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