AN ADDITIONAL GIBBS’ STATE FOR THE CUBIC

SCHRÖDINGER EQUATION ON THE CIRCLE

K.L. Vaninsky

Courant Institute 251 Mercer Street New York University New York, NY 10012

Abstract. An invariant Gibbs’ state for the nonlinear Schrödinger equation on the circle was constructed by Bourgain, [B], and McKean, [MC], out of the basic Hamiltonian using a trigonometric cut-off. The cubic nonlinear Schrödinger equation is a completely integrable system having an infinite number of additional integrals of motion. In this paper we construct the second invariant Gibbs’ state from one of these additional integrals for the cubic NLS on the circle. This additional Gibbs’ state is singular with respect to the Gibbs’ state previously constructed from the basic Hamiltonian. Our approach employs the Ablowitz-Ladik system, a completely integrable discretization of the cubic Schrödinger equation.

1. Introduction. The nonlinear Schrödinger equation for the complex function ψ(x, t), x ∈ 𝕊, t ∈ R^{1} is

iψ^{•} = −ψ′′ + p|ψ|^{2p−2}ψ,

where p is an arbitrary integer. It can be written in Hamiltonian form. Let M = (ψ, ψ̄) be a space of pairs of two complex functions ψ = Q + iP and ψ̄ = Q − iP on the circle 𝕊 of perimeter 1. For any two functionals F and G on M, define

\{F, G\}_{ω} = i ∫_{𝕊} \frac{δF}{δψ} \frac{δG}{δψ̄} − \frac{δF}{δψ̄} \frac{δG}{δψ}dx.

2000 Mathematics Subject Classification. 58F07, 70H15. Key words and phrases. Spectral curve. Moduli. Symplectic volume. The work is partially supported by NSF grant DMS-9501002 and DMS-9971834

Typeset by AMS-TEX

The Hamiltonian H(ψ, ψ̄) = ∫_{𝕊} |ψ′|^{2} + |ψ|^{2p}dx produces the nonlinear Schrö - dinger flow

ψ^{•} = {ψ, H}_{ω}

The equation has two other generic integrals. They are N = ∫ |ψ|^{2}dx, the number of particles, and P = − ∫ iψ′ψ̄ dx, the momentum.

The Gibbs’ measure associated with the basic Hamiltonian

e^{− \frac{1}{2} H}d vol = e^{− \frac{1}{2} H} \frac{1}{∞!} ⋀_{∞} ω = e^{− \frac{1}{2} H} \frac{1}{∞!} ⋀_{x∈𝕊} iδψ(x) ∧ δψ̄(x)

was constructed by Bourgain and McKean [B1-2, MC]; it is the product of two independent copies of circular Brownian motion for the components of the function ψ = Q + iP coupled together by the nonlinear factor e^{− \frac{1}{2} ∫_{𝕊} |Q^{2}+P^{2}|^{p}dx}. This measure is invariant under the flow; for any p the later exists almost everywhere with respect to the Gibbs’ measure.

The cubic NLS corresponds to the case p = 2. It has infinite series of commuting Hamiltonians. The first five are listed below

H_{1} = ∫ |ψ|^{2}dx,
H_{2} = ∫ −iψ′ψ̄dx,
H_{3} = ∫ |ψ′|^{2} + |ψ|^{4}dx,
H_{4} = ∫ iψ′′′ψ̄ − iψ̄′ψ|ψ|^{2} − 4|ψ|^{2}ψ̄ψ′dx,
H_{5} = ∫ |ψ′′|^{2} + 2|ψ|^{6} + 8|ψ′|^{2}|ψ|^{2} + ψ̄′^{2}ψ^{2} + ψ′^{2}ψ̄^{2}dx.

The classical generic integrals N , P and H are the first three of this infinite series. All of them are integrals of the isobaric polynomials, counting ψ and d/dx as having degree 1; for example, H_{5} is an integral of an isobaric polynomial of degree 6.

We consider the phase space M of pairs (ψ, ψ̄) from the Sobolev space H^{1}*. The original equation defines a global flow on M, [B2]. In this paper we construct an invariant Gibbs’ state from the higher Hamiltonian H_{5}:

e^{− \frac{1}{2} H_{5}}d vol = e^{− \frac{1}{2} H_{5}} \frac{1}{∞!} ⋀_{x∈𝕊} iδψ(x) ∧ δψ̄(x).

*If ψ(x) = ∑_{k} e^{2πikx}ψ̂(k), then the Sobolev norm of ψ(x) is |ψ|_{H^{s}}^{2} = ∑_{k} |ψ̂(k)|^{2}(1 + k^{2})^{s}.

This state is singular with respect to the Gibbs’ state e^{− \frac{1}{2} H_{3}}d vol. To construct the measure and prove its invariance we will use the Ablowitz-Ladik (AL) system.

The AL equation for the complex N-periodic function ψ(n, t) , n ∈ Z^{1}, t ∈ R^{1}, is*

iψ^{•}_{n} = −(ψ_{n+1} + ψ_{n−1} − 2ψ_{n}) + |ψ_{n}|^{2}(ψ_{n+1} + ψ_{n−1}).

It is easy to check that the quantity

D = ∏_{k=0}^{N−1} R_{k} = ∏_{k=0}^{N−1}(1 − |ψ_{k}|^{2})

is an integral of motion. We assume that |ψ_{n}| < 1 for all n. This area of the phase space we call the box B; it is invariant under the flow. In the box all quantities R_{n} and D are positive.

In fact, AL has many integrals of motion. The first three interesting ones are I_{0}, I_{2}, I_{4}:

NI_{0} ≡ −\frac{1}{2} log D,
NI_{2} = ∑_{r=0}^{N−1} ψ_{r}\overline{ψ}_{r−1},

NI_{4} = ∑_{r=0}^{N−1} ψ_{r}\overline{ψ}_{r−2} (1 − |ψ_{r−1}|^{2}) − \frac{1}{2}(ψ_{r}\overline{ψ}_{r−1})^{2}.

From these integrals we form Hamiltonians

H_{1} = N(I_{0} + Ī_{0})
H_{3} = N(I_{2} + Ī_{2} − 2I_{0} − 2Ī_{0})
H_{5} = N(I_{4} + Ī_{4} − 4I_{2} − 4Ī_{2} + 6I_{0} + 6Ī_{0}).

Let M_{N} be a space of complex N-periodic sequences ψ_{n} = Q_{n} + iP_{n}, n ∈ Z^{1} with |ψ_{n}| < 1. Introducing the bracket

\{f, g\}_{ω_{0}} ≡ i ∑_{n=1}^{N} R_{n} (\frac{δf}{δψ_{n}} \frac{δg}{δψ̄_{n}} − \frac{δf}{δψ̄_{n}} \frac{δg}{δψ_{n}} ) ,

for two functionals f and g on M_{N} we can write the original AL flow in Hamiltonian form

ψ^{•}_{n} = {ψ_{n}, H_{3}}_{ω_{0}}.

*• denotes a time derivative; ψ_{n} = ψ(n, t); periodicity means ψ_{n+N} = ψ_{n} for n ∈ ℤ.

The symplectic form ω_{0} producies the volume element

d vol = \frac{1}{N!D_{N}} ⋀_{n} iδψ̄_{n} ∧ δψ_{n}.

The volume of the box is infinite

∫_{B} d vol = ∫_{B} \frac{1}{N!D_{N}} ⋀_{n} iδψ̄_{n} ∧ δψ_{n} = ∞

due to the singularity of the factor \frac{1}{D_{N}}. On the box we can define the Gibbs’ state with the density

e^{− \frac{N^{5}}{2} H_{5}}d vol = e^{− \frac{N^{5}}{2} H_{5}} \frac{1}{N!} ⋀_{n} ω_{0} = e^{− \frac{N^{5}}{2} H_{5}} \frac{1}{N!D_{N}} ⋀_{n} iδψ_{n} ∧ δψ̄_{n},

The singularity of the volume element near the boundary is rectified by the vanishing factor e^{− \frac{N^{5}}{2} 12NI_{0}} = D^{3N^{5}}: the total mass

∫_{B} e^{− \frac{N^{5}}{2} H_{5}}d vol < ∞.

The area of the box close to the boundary has negligible probability. The measure of the area outside the box we take to be zero.

Let ϵ = \frac{1}{N}. Under the scaling ψ_{n} = ϵψ(\frac{n}{N}) and θ = ϵ^{−2}t, the AL vector field approaches the NLS vector field as N → ∞. Indeed,

ψ_{n+1} = ψ_{n} + ψ′_{n}ϵ + \frac{1}{2}ψ′′_{n}ϵ^{2} …, ψ_{n−1} = ψ_{n} − ψ′_{n}ϵ + \frac{1}{2}ψ′′_{n}ϵ^{2} ….

So,

ψ_{n+1} + ψ_{n−1} − 2ψ_{n} = ψ′′_{n}ϵ^{2} + ….

and the flow in the time scale θ,

i\frac{∂ψ_{n}}{∂θ} = −(ψ_{n+1} + ψ_{n−1} − 2ψ_{n}) + |ψ_{n}|^{2}(ψ_{n+1} + ψ_{n−1}),

scales as follows:

iϵ^{3} \frac{∂ψ}{∂t} = −ϵψ′′ϵ^{2} + ϵ^{3}|ψ|^{2}2ψ + ….

This indicates that AL flow converges to the NLS flow, though statement can not be taken literaly because these two flows live in two different spaces M_{N} and M.

The integrals H_{1}, H_{3} and H_{5} of the AL system converge to H_{1}, H_{3} and H_{5}:

N H_{1}(ψ_{n}, ψ̄_{n}) → H_{1}(ψ, ψ̄),
−N^{3}H_{3}(ψ_{n}, ψ̄_{n}) → H_{3}(ψ, ψ̄),
N^{5}H_{5}(ψ_{n}, ψ̄_{n}) → H_{5}(ψ, ψ̄).

Also, for any ψ, we have D_{N}(ψ_{n}, ψ̄_{n}) → 1, as N → ∞. Therefore, one should expect to get from the Gibbs’ state for AL, in the limit of N → ∞, the desired invariant measure

e^{− \frac{1}{2} H_{5}}d vol = e^{− \frac{1}{2} H_{5}} \frac{1}{∞!} ⋀_{x∈𝕊} iδψ(x) ∧ δψ̄(x).

Again, this statement is correct but can not be taken literaly, because these measures do not live on the same space. To make the arguments rigorous we need to embed the AL flows and measures into the function space. For this, we use interpolating trigonometrical polynomials.

The main goal of the paper is the space-time random field ψ(x, t), x ∈ 𝕊, t ∈ R^{1} such that:

(i) ψ(x, t) is stationary respect x and t;

(ii) ψ(•, t) has Gibbs’ distribution e^{− \frac{1}{2} H_{5}}d vol;

(iii) the random variable ψ(•, t), t ≠ 0 is measurable with respect to the σ-field generated by ψ(•, 0); the measure is supported by the solutions of NLS;

(iv) The x-derivative of almost every realisation of the random field is Hölder continuous:

|∂_{x}ψ(x_{1}, t_{1}) − ∂_{x}ψ(x_{2}, t_{2})| ≤ K [|x_{1} − x_{2}|^{\frac{1}{2} −} + |t_{1} − t_{2}|^{\frac{1}{4} −}]

with a random constant K. The exponents \frac{1}{2} and \frac{1}{4} are optimal.

The paper is organised as follows. In the section 2 we introduce the commuting flows of the AL hierarchy and commuatator formulas for them. Section 3 is devoted to the study of the direct spectral problem for the auxiliary linear system. We explain the details of the spectral curve for various types of potentials. Invariant quantities are computed in section 4. The Floquet and dual Floquet solutions are studied in sections 5 and 6. Section 7 provides formulas for the symplectic structure and the Poisson bracket. This completes the first part of the paper about integrability properties and Hamiltonian formalism for the AL system.

The second part starts with section 8 where the strategy for constructing the measure is outlined; it is implemented in the subsequent sections 9-11.

Finally I would like to thank I. Krichever, H. McKean, V. Peller and J. Zubelli for helpful discussion. It is also pleasure to thank MPI in Bonn and IMPA for their hospitality.

2. Ablowitz–Ladik hierarchy. The Ablowitz–Ladik equation for the complex N–periodic function ψ(n, t) n ∈ Z^{1}, t ∈ R^{1} is

(1)  iψ^{•}_{n} = −(ψ_{n+1} + ψ_{n−1} − 2ψ_{n}) + |ψ_{n}|^{2}(ψ_{n+1} + ψ_{n−1}).

Recall that

D = ∏_{k=0}^{N−1} R_{k} = ∏_{k=0}^{N−1}(1 − |ψ_{k}|^{2})

is an integral of motion, whence the ”box” B = (ψ : |ψ_{n}| < 1 for all n) is invariant under the flow. In the box R_{n} and D are positive.

The flow (1) is one of infinitely many flows of the AL hierarchy. The rotation or the phase flow

(2)  iψ^{•}_{n} = −ψ_{n}

is the first flow of the hierarchy. Let

V_{2}(n, t, λ) = \frac{1}{\sqrt{R_{n}}} [ \begin{matrix}λ & ψ_{n} \\ \overline{ψ}_{n} & λ^{−1} \end{matrix} ] .

The phase flow is the compatibility condition for

(3)  [∂_{t} − V_{1}, Δ − V_{2}] = 0,

where V_{1} = iσ_{3}/2 and Δf_{n} = f_{n+1} is a shift operator. Here and below σ denotes the Pauli matricies

σ_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , σ_{2} = ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) , σ_{3} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

The original AL equation (1) is the compatibility condition for*

(4)  [∂_{t} − V_{3}, Δ − V_{2}] = 0,

where

V_{3}(n, t, λ)

= i [ \begin{matrix}λ^{2} − 1 + \frac{3}{2}ψ_{n}\overline{ψ}_{n−1} − \frac{1}{2} \overline{ψ}_{n}ψ_{n−1} & ψ_{n}λ − ψ_{n−1}λ^{−1} \\ \overline{ψ}_{n−1}λ − \overline{ψ}_{n}λ^{−1} & 1 − λ^{−2} + \frac{1}{2} \overline{ψ}_{n}ψ_{n−1} + \frac{1}{2}ψ_{n}\overline{ψ}_{n−1} \end{matrix} ] .

The formula means that**

(∂_{t} − V_{3}(n + 1))(Δ − V_{2}(n)) − (Δ − V_{2}(n))(∂_{t} − V_{3}(n)) = 0

i.e.

V^{•} _{2}(n) = V_{3}(n + 1)V_{2}(n) − V_{2}(n)V_{3}(n).

*A similar form of the commutator formalism was considered in [AL, MEKL]. Our form has the small advantage, that it leads to a unimodular monodromy matrix.

**The operator V = V_{2} acts like shift: V_{2}(n)f_{n} = f_{n+1}.

3. Monodromy matrix. The spectral curve. Since ψ_{n+N} = ψ_{n}, all matricies V_{k}, k = 1, 2, …, satisfy the periodicity condition V_{k}(n + N) = V_{k}(n). We introduce the transition matrix

T_{n,m}(t, λ) ≡ V (n − 1, t, λ)V (n − 2, t, λ) … V (m, t, λ) (n > m).

It is easy to see that the spectrum of T_{m+N,m}(t, λ) does not depend on m. Indeed,

T_{m+1+N,m+1}(t, λ) = V (m + N) … V (m + 1) = V (m + N)T_{m+N,m}V (m)^{−1}.

Also, the spectrum of T_{N} ≡ T_{N,0} does not depend on time. This follows from

the identity T^{•}_{N} = [V_{3}(0), T_{N}], which is proved as follows:

∂_{t}[V … V (k + 1)V (k) … V]

= … + V … V^{•} (k + 1)V (k) … V + V … V (k + 1)V^{•} (k) … V + …

= … + V … (V_{3}(k + 2)V (k + 1) − V (k + 1)V_{3}(k + 1))V (k) … V

+ V … V (k + 1)(V_{3}(k + 1)V (k) − V (k)V_{3}(k)) … V + …

= V_{3}(N)T_{N}(λ) − T_{N}(λ)V_{3}(0).

Consider the special ”Floquet” solution

g_{n} = [ \begin{matrix}g_{n}^{1}(λ) \\ g_{n}^{2}(λ) \end{matrix} ]

of the eigenvalue problem g_{n+1}(λ) = V (n, λ)g_{n}(λ) specified by the condition

g_{N}(λ) = T_{N}g_{0}(λ) = wg_{0}(λ),

where w being the complex ”multiplier” determined by

0 = det [ \begin{matrix}T_{N}^{11} − w & T_{N}^{12} \\ T_{N}^{21} & T_{N}^{22} − w \end{matrix} ] = w^{2} − 2wΔ(λ) + 1,

where Δ(λ) = tr T_{N}(λ)/2. We know that Δ(λ) is an integral of motion. The multiplier is

w = Δ(λ) + \sqrt{Δ^{2}(λ) − 1};
It  becomes  single-valued  on  the  spectral  curve

Γ = {Q = (λ, y) ∈ ℂ^{2} : y^{2} = Δ^{2}(λ) − 1}.

The curve Γ inherits symmetries from V_{2}(n, λ). It is easy to check that

(1)  σ_{1}V (\frac{1}{\overline{λ}}) σ_{1} = \overline{V (λ)},
(2)  σ_{3}V (−λ)σ_{3} = −V (λ).

Obviously T_{N}(λ) satisfies similar identities. Whence

(3)  Δ (\frac{1}{\overline{λ}}) = \overline{Δ(λ)},
(4)  Δ(−λ) = (−1)^{N}Δ(λ).

The symmetries (3-4) produce an antiholomorhic involution τ_{a} and a holomorphic involution τ on the curve

τ_{a} : (λ, y) ⟶ (\frac{1}{\overline{λ}}, \overline{y}) , (5)

τ : (λ, y) ⟶ (−λ, (−1)^{N}y) . (6) On Γ, there exists another holomorphic involution τ_{±} which permutes the sheets, viz. τ_{±} : (λ, y) → (λ, −y) . From the quadratic equation for w, we have w(Q)w(τ_{±}Q) = 1. Remark. Obviously, we have freedom in the choice of sign for the second coordinate of the involution, say

τ_{a} : (λ, y) ⟶ (\frac{1}{\overline{λ}}, ±\overline{y}) .

The sign in (5–6) is chosen in such way that τ_{a} and τ preserve the infinities of the curve (see example 5 of this section). Example 1: vanishing potential ψ_{n} ≡ 0. Then

T_{N}(λ) = ( \begin{matrix}λ^{N} & 0 \\ 0 & λ^{−N} \end{matrix} ) , Δ(λ) = cosh N log λ and w(Q) = e^{±N log λ(Q)}.

The points λ_{k}^{±} = e^{i \frac{2π}{2N} k}, k = 0, ⋯, 2N − 1 satisfy the condition Δ^{2}(λ_{k}^{±}) = 1 and at these points* Δ^{•}(λ_{k}^{±}) = 0. The points λ_{k}^{±} are simple crossings of the curve Γ. They form periodic/antiperiodic spectrum; namely, w = +1 for k even and w = −1 for k odd. If λ = e^{iθ}, then Δ(e^{iθ}) = (e^{iθN} + e^{−iθN}) /2 = cos θN. The graph of Δ(λ) for N = 2 is shown in fig 1.

When ψ_{n} is not identicaly zero, then the double points λ_{k}^{±} split into pairs λ_{k}^{−} and λ_{k}^{+} of simple ramification. The symmetry (3) implies that Δ(λ) is always real for |λ| = 1. Note also that, due to the symmetry (4), the branch points form symmetric pairs λ_{k}^{−}, λ_{k}^{+} and i λ_{k′}^{−}, λ_{k′}^{+} such that λ_{k}^{±} = −λ_{k′}^{±}, when k − k′ ≡ 0 (mod N).

The branch points λ_{k}^{±} lie only on the unit circle**. Indeed, the equation

*• now denotes derivative in λ variable.

**The statement is proved by the method of [AL].

fig. 1

g_{n+1}(λ) = V (n, λ)g_{n}(λ) can be written in the form Λ_{n}g_{n}(λ) = λg_{n}(λ), where

Λ_{n} = [ \begin{matrix}\sqrt{R_{n}}Δ & −ψ_{n} \\ \overline{ψ}_{n−1} & \sqrt{R_{n−1}}Δ^{−1} \end{matrix} ] and Δf_{n} = f_{n+1}.

It is easy to see that Λ is unitary in the space of 2N–periodic vector functions with the complex inner product < f, g >_{2♡}= \frac{1}{2N} ∑_{k=0}^{2N−1} f_{k}\overline{g}_{k}. Indeed, introducing the formal inverse

Λ_{n}^{−1} = [ \begin{matrix}\sqrt{R_{n−1}}Δ^{−1} & ψ_{n−1} \\ −\overline{ψ}_{n} & \sqrt{R_{n}}Δ \end{matrix} ]

it is easy to derive the Cauchy identity

< Λf, g >_{♡} =< f, Λ^{−1}g >_{♡} +

+ \frac{1}{N} \sqrt{R_{N−1}} (f_{N}^{1} \overline{g}_{N−1}^{1} − f_{0}^{1}\overline{g}_{−1}^{1}) + \frac{1}{N} \sqrt{R_{N−1}} (f_{−1}^{2}\overline{g}_{0}^{2} − f_{N−1}^{2}\overline{g}_{N}^{2}) , (7)

where < f, g >_{♡}= \frac{1}{N} ∑_{k=0}^{N−1} f_{k}\overline{g}_{k}. Appling this formula twice, first to the interval k = 0, ⋯, N − 1 and then to the interval k = N, ⋯, 2N − 1 we obtain a similar identity for < •, • >_{2♡} which implies the result. Example 2: N = 2–periodic case. It is instructive to analyze this case complitely. The potential can be written in the form

ψ_{n} = \{ \begin{matrix}A + B, & n is even \\ A − B, & n is odd. \end{matrix}

The matrix T_{2} can be easily computed:

T_{2} = \frac{1}{\sqrt{D}} [ \begin{matrix}λ^{2} + (A − B)(\overline{A} + \overline{B}) & λ(A + B) + λ^{−1}(A − B) \\ λ(\overline{A} − \overline{B}) + λ^{−1}(\overline{A} + \overline{B}) & λ^{−2} + (A + B)(\overline{A} − \overline{B}) \end{matrix} ] .

Then Δ(θ) = \frac{1}{\sqrt{D}} [cos 2θ + |A|^{2} − |B|^{2}] for λ = e^{iθ}, and D = (1−|A+B|^{2}) ×(1− |A−B|^{2}). Consider the case B ≡ 0 and A ≠ 0. Then Δ(θ) = \frac{1}{\sqrt{D}} [cos 2θ + |A|^{2}] , and D = (1 − |A|^{2})^{2}. The graph of Δ(θ) is shown in fig. 2.

fig. 2

The double points λ_{0}^{±} and λ_{2}^{±} split into pairs, while λ_{1}^{±} and λ_{3}^{±} remain double.

Now consider the case when A = 0 and B ≠ 0. Then, Δ(θ) = \frac{1}{\sqrt{D}} [cos 2θ −|B|^{2}], D = (1 − |B|^{2})^{2}. In this case, the double points λ_{0}^{±} and λ_{2}^{±} do not split, while λ_{1}^{±} and λ_{3}^{±} split into pairs of simple roots.

Finally, there is an open area in the space of parameters, say, A ∼ B, where

Δ(θ) ≈ \frac{1}{\sqrt{D}} cos 2θ, D ≈ 1 − 4|A|^{2} < 1.

In this case all double roots split into pairs of simple roots. Example 3: constant potential ψ_{n} = ψ_{0}. Consider, first, the case N = 2. Then

T_{2} = \frac{1}{\sqrt{D}} [ \begin{matrix}λ^{2} + |ψ_{0}|^{2} & λψ_{0} + λ^{−1}ψ_{0} \\ λ\overline{ψ}_{0} + λ^{−1}\overline{ψ}_{0} & λ^{−2} + |ψ_{0}|^{2} \end{matrix} ] ,

Δ(θ) = \frac{1}{\sqrt{D}} [cos 2θ + |ψ_{0}|^{2}] , and D = (1 − |ψ_{0}|^{2})^{2}. The branch points λ_{k}^{±} are given by the equation Δ^{2}(λ) = 1. Obviously, the points λ_{1}^{±} and λ_{3}^{±} do not split. For λ_{0}^{±} we have the equation

1 − |ψ_{0}|^{2} = cos 2θ + |ψ_{0}|^{2},

so for small |ψ_{0}|,

1 − 2|ψ_{0}|^{2} = 1 − 2θ^{2} + O(θ^{4}).

Finally, we arrive at θ = |ψ_{0}| + O(|ψ_{0}|^{3}) and

λ_{0}^{±} = e^{±i(|ψ_{0}|+O(|ψ_{0}|^{3}))} λ_{2}^{±} = e^{±i(|ψ_{0}|+O(|ψ_{0}|^{3}))+iπ}.

In words, the open gap is proportional to the absolute value of the potential |ψ_{0}|.

The case of general N can be treated easily. One has to compute spectrum of T_{N}. After simple algebra

λ_{0}^{±} = e^{±i(|ψ_{0}|+O(|ψ_{0}|^{3}))}, λ_{N}^{±} = e^{±i(|ψ_{0}|+O(|ψ_{0}|^{3}))+iπ}.

Example 4: two-gap trigonometric potential. Consider the N–periodic potential ψ^{k}:

ψ_{n}^{k} = e^{iϕ_{k}n}ψ_{0}, ϕ_{k} = \frac{2πk}{N} k = 0, ⋯, N − 1.

We will show that this potential opens k–th and k + N–th gap as in

(8)  λ_{k}^{±} = e^{iϕ_{k}/2±i(|ψ_{0}|+O(|ψ_{0}|^{3}))},

and

(9)  λ_{k+N}^{±} = e^{iϕ_{k}/2±i(|ψ_{0}|+O(|ψ_{0}|^{3}))+iπ}.

To emphasise the dependence on ψ^{k} we write V (n, λ|ψ^{k}) = V^{k}(n), T (λ|ψ^{k}) = T^{k}(λ) and Δ(λ|ψ^{k}) = Δ^{k}(λ). Note, first, that

L^{k}(n) = Φ^{n}V^{k}(0)Φ^{−n}, where Φ = e^{iσ_{3}ϕ_{k}/2}.

Therefore,

T (λ|e^{iϕ_{k}n}ψ_{0}) = V^{k}(N − 1) ⋯ V^{k}(0) = Φ^{N}(Φ^{−1}V^{k}(0))^{N}
= (−1)^{k}(Φ^{−1}V^{k}(0))^{N} = (−1)^{k}T (λe^{−iϕ_{k}/2}|e^{−iϕ_{k}/2}ψ_{0}).

Taking the trace we obtain

Δ(θ|e^{iϕ_{k}n}ψ_{0}) = (−1)^{k}Δ(θ − \frac{ϕ_{k}}{2} |e^{−iϕ_{k}/2}ψ_{0}).

This and the result of Example 3 produce the formula (8). (9) follows from this and the symmetry (4). Remark 1. The situation here is similar to the NLS equation, [MCV1], where the N–th gap opens in proportion to the N-th Fourier coefficient. Remark 2. The 4N branch points λ_{0}^{±}, ⋯, λ_{2N−1}^{±} determine the curve. The symmetry (4) leaves only 2N free parameters. In fact, there are only N free independent parameters due to periodicity conditions. Example 5: generic periodic potential. The potential can be written as a sum of harmonics ψ^{k}, k = 0, ⋯, N − 1. All gaps are open. To construct the curve Γ, let us take two copies “+” and “-” of ℂP^{1} cut along circular arcs connecting λ_{k}^{−} and λ_{k}^{+} (fig. 3)

fig. 3

Each copy ℂP^{1} has two marked points P_{0/∞}^{+} and P_{0/∞}^{−}. The behavior of Δ(λ), y(Q) = \sqrt{Δ(λ)^{2} − 1}, and w(Q) near these points is this:

Δ(λ) ∼ \frac{1}{2}λ^{N}, y(Q) ∼ ±\frac{1}{2}λ^{N}, w(Q) ∼ λ^{±N}, λ = λ(Q), Q ∈ (P_{∞}^{±});

and

Δ(λ) ∼ \frac{1}{2}λ^{−N}, y(Q) ∼ ±\frac{1}{2}λ^{−N}, w(Q) ∼ λ^{∓N}, λ = λ(Q), Q ∈ (P_{0}^{±}).

From this, we see that τ_{a} maps point Q = (λ, y) in the vicinity of P_{∞}^{+/−} to the point τ_{a}Q = (\frac{1}{\overline{λ}}, \overline{y}) in the vicinity of P_{0}^{+/−}. The curve is obtained by gluing

fig. 4

together “±” copies of ℂP^{1} along the cuts and identifying the points P_{∞}^{+/−} with P_{0}^{+/−} as specified

P^{+} = P_{∞}^{+} + P_{0}^{+},  and  P^{−} = P_{∞}^{−} + P_{0}^{−}.

The cuts are the projections on the λ–plane of the real ovals of anti–holomorphic involution τ_{a}. The infinities P^{+} and P^{−} are also fixed points of τ_{a}. The multiplier w(Q) is single-valued on Γ and satisfies the identity w(τ_{a}Q) = \overline{w(Q)}. This guaranties continuity of w(Q) at the points P^{+/−}, where w(Q) ⟶ ∞/0 respectively. The involution τ is defined so that it preserves the infinities. The formula w(Q) = Δ(λ(Q)) + y(Q) implies w(τQ) = (−1)^{N}w(Q).

Let us introduce a multivalued function p(Q) on Γ by the formula w(Q) = e^{p(Q)N}. Obviously, p(Q) is defined up to the integer multiple of 2πi/N. The differential dp is of the third kind with poles at P^{+} and P^{−}. The relation w(Q)w(τ_{±}Q) = 1 implies that p(Q) ≡ −p(τ_{±}Q) (mod 2πi/N). Therefore, in the vicinity of P_{∞}^{±},

p_{∞}^{+}(λ) = + log λ + I_{0} + \frac{I_{1}}{λ} + \frac{I_{2}}{λ} + ⋯,
p_{∞}^{−}(λ) = − log λ − I_{0} − \frac{I_{1}}{λ} − \frac{I_{2}}{λ} + ⋯,

where λ = λ(Q). The fact that τ preserves P_{∞}^{+/−} and w(τQ) = (−1)^{N}w(Q) implies I_{k} = 0 for k odd, so

p_{∞}^{+}(λ) = + log λ + I_{0} + \frac{I_{2}}{λ^{2}} + \frac{I_{4}}{λ^{4}} + ⋯,
p_{∞}^{−}(λ) = − log λ − I_{0} − \frac{I_{2}}{λ^{2}} − \frac{I_{4}}{λ^{4}} + · · · .

The anti-involution τ_{a} maps P_{∞}^{+/−} to P_{0}^{+/−}. This and relation w(τ_{a}Q) = \overline{w(Q)} imply, in the vicinity of P_{0}^{±},

p_{0}^{+}(λ) = − log λ + \overline{I}_{0} + \overline{I}_{2}λ^{2} + \overline{I}_{4}λ^{4} + ⋯,
p_{0}^{−}(λ) = + log λ − \overline{I}_{0} − \overline{I}_{2}λ^{2} − \overline{I}_{4}λ^{4} + · · · .

The explicit form of the integrals I_{0}, I_{2}, ⋯ will be computed in the next section.

4. Integrals of motion. To simplify the calculations, we introduce V_{new}(n) = \sqrt{R_{n}}V_{old}(n). The corresponding mulitipliers are w_{new} = \sqrt{D}w_{old}, and

p_{old} = p_{new} − \frac{1}{2N} log D.

We write all formulas in this section for the new spectral problem.

We introduce T_{n,m} = (1+W_{n}(λ))e^{Z_{n,m}(λ)}(1+W_{m}(λ))^{−1}, where Z is a diagonal matrix and W is antidiagonal. Separating the diagonal and antidiagonal part in the equation T_{n+1,m} = V (n)T_{n,m}, we obtain

e^{Z_{n+1,m}−Z_{n,m}} = Λ + Ψ_{n}W_{n}, (1)

W_{n+1}e^{Z_{n+1,m}−Z_{n,m}} = ΛW_{n} + Ψ_{n}, (2)

where

Λ = [ \begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix} ] and Ψ_{n} = [ \begin{matrix}0 & ψ_{n} \\ \overline{ψ}_{n} & 0 \end{matrix} ] .

Let W_{n}(λ) = ∑_{k=1}^{∞} W_{n}^{k}λ^{−k}, where

W_{n}^{k} = [ \begin{matrix}0 & w_{n}^{k+} \\ w_{n}^{k−} & 0 \end{matrix} ] .

From (1-2), we have W_{n+1}(Λ + Ψ_{n}W_{n}) = ΛW_{n} + Ψ_{n}. Introducing

σ_{+} = [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] and σ_{−} = [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ] ,

we can write Λ = σ_{+}λ + σ_{−}λ^{−1} and

(3)  \begin{aligned}∑_{k≥1} W_{n+1}^{k}λ^{−k}(λσ_{+} & + λ^{−1}σ_{−}) + ∑_{k,p≥1} W_{n+1}^{r}Ψ_{n}W_{n}^{p}λ^{−r−p} \\ & = (λσ_{+} + λ^{−1}σ_{−}) ∑_{k≥1} W_{n}^{k}λ^{−k} + Ψ_{n}. \end{aligned}

Collecting terms with λ^{0} in (3), we have W_{n+1}^{1}σ_{+} = σ_{+}W_{n}^{1} + Ψ_{n} and

W_{n}^{1} = [ \begin{matrix}0 & −ψ_{n} \\ \overline{ψ}_{n−1} & 0 \end{matrix} ] .

Collecting terms with λ^{−1} in (3), we have W_{n+1}^{2}σ_{+} = σ_{+}W_{n}^{2}. This implies W_{n}^{2} ≡ 0.

Collecting terms in (3) of the order λ^{−k}, k ≥ 2, we have the reccurence formula

(4) W_{n+1}^{k+1}σ_{+} − σ_{+}W_{n}^{k+1} = σ_{−}W_{n}^{k−1} − W_{n+1}^{k−1}σ_{−} − ∑_{\begin{matrix}r+p=k, \\ r,p≥1 \end{matrix}} W_{n+1}^{r}Ψ_{n}W_{n}^{p};

whence, for k = 2,

W_{n}^{3} = [ \begin{matrix}0 & −ψ_{n+1} + ψ_{n+1}|ψ_{n}|^{2} \\ \overline{ψ}_{n−2} − \overline{ψ}_{n−2}|ψ_{n−1}|^{2} & 0 \end{matrix} ] .

Therefore, for n = N and m = 0 we have

T_{N}(λ) = (1 + W_{N}(λ))e^{Z_{N,0}(λ)}(1 + W_{0}(λ))^{−1}

and

e^{Z_{N,0}}(λ) = ∏_{r=0}^{N−1}(Λ + Ψ_{r}W_{r}).

Using the fact that Ψ_{r}W_{r} is diagonal we infer that for Q ∈ (P_{∞}^{+})

w(Q) = e^{p_{∞}^{+}(λ)N} = ∏_{r=0}^{N−1} (λ + ψ_{r} ∑_{k≥1} w_{r}^{k−}λ^{−k}) .

Therefore,

p_{∞}^{+}(λ)N = N log λ+

+ \frac{1}{λ^{2}} ∑_{r=0}^{N−1} ψ_{r}w_{r}^{1−} + \frac{1}{λ^{4}} ( ∑_{r=0}^{N−1} ψ_{r}w_{r}^{3−} − \frac{1}{2}(ψ_{r}w_{r}^{1−})^{2}) + · · · .

Using the explicit expression of W_{n}^{1}, W_{n}^{3}, we have*

NI_{0} ≡ −\frac{1}{2} log D,
NI_{2} = ∑_{r=0}^{N−1} ψ_{r}\overline{ψ}_{r−1},

NI_{4} = ∑_{r=0}^{N−1} ψ_{r}\overline{ψ}_{r−2} (1 − |ψ_{r−1}|^{2}) − \frac{1}{2}(ψ_{r}\overline{ψ}_{r−1})^{2}.

5. Floquet solution. Baker–Akhiezer function. The Floquet solution was defined in section 3 as a special solution of the spectral problem g_{n+1} = V (n)g_{n} with the property g_{N} = T_{N}g_{0} = wg_{0}. It is uniqlue specified by the boundary conditions g_{n}^{1} + g_{n}^{2}|_{n=0} = 1. Example: vanishing potential ψ_{n} = 0. Then,

V_{2}(n, λ) = [ \begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix} ] ,  and  T_{N}(λ) = [ \begin{matrix}λ^{N} & 0 \\ 0 & λ^{−N} \end{matrix} ] .

Therefore,** for Q ∈ (P_{∞}^{+}/P_{0}^{+}),

g_{n}(Q) = e^{±n log λ} [g_{0}^{+}/ĝ_{0}^{+}] , w(Q) = e^{±N log λ};

and for Q ∈ (P_{∞}^{−}/P_{0}^{−})

g_{n}(Q) = e^{∓n log λ} [g_{0}^{−}/ĝ_{0}^{−}] , w(Q) = e^{∓N log λ};

*We write the coefficients for the old spectral problem.

**For any g = [ \begin{matrix}a \\ c \end{matrix} ] we write ĝ = [ c̄ ; ā ].

vectors g_{0}^{+}, g_{0}^{−} are

g_{0}^{+} = [ \begin{matrix}1 \\ 0 \end{matrix} ] , g_{0}^{−} = [ \begin{matrix}0 \\ 1 \end{matrix} ] .

We introduce another Floquet solution g̃_{n} normalised by the condition ig̃_{n}^{1} − ig̃_{n}^{2}|_{n=0} = 1. Let us define

D_{0} = 1,  D_{n} = ∏_{k=0}^{n−1} R_{k}, n = 1, …, N − 1;  D_{N} = D.

Lemma 1. i. The Floquet solution g_{n}(Q) is:

g_{n}(Q) = C(Q) [ \begin{matrix}T_{n}^{11}(λ) \\ T_{n}^{21}(λ) \end{matrix} ] + (1 − C(Q)) [ \begin{matrix}T_{n}^{12}(λ) \\ T_{n}^{22}(λ) \end{matrix} ] , λ = λ(Q).

with

C(Q) = \frac{T^{12}}{T^{12} − T^{11} + w(Q)} = \frac{w(Q) − T^{22}}{T^{21} − T^{22} + w(Q)},

here, T = T_{N}.

ii. The Floquet solution g̃_{n}(Q) is

g̃_{n}(Q) = R(Q) [ \begin{matrix}T_{n}^{11}(λ) \\ T_{n}^{21}(λ) \end{matrix} ] + (i + R(Q)) [ \begin{matrix}T_{n}^{12}(λ) \\ T_{n}^{22}(λ) \end{matrix} ] ,

with

R(Q) = \frac{iT^{12}}{w(Q) − T^{11} − T^{12}} = \frac{iT^{22} − iw(Q)}{w(Q) − T^{21} − T^{22}} .

iii. The following formulas hold:

σ_{1}g_{n}(τ_{a}Q) = ḡ_{n}(Q), σ_{1}g̃_{n}(Q)(τ_{a}Q) = \overline{g̃_{n}(Q)}.

and

g_{n}(τQ) = (−1)^{n}iσ_{3}g̃_{n}(Q).

(iv). The Floquet solution g_{n}(Q) has 2N poles at the points γ’s on the real ovals of the curve Γ. On each oval there is just one γ.

(v). If Q ∈ (P_{∞}^{+}/P_{0}^{+}), then g_{n}(Q) has the development

g_{n}(Q) = e^{±n log λ} \frac{1}{\sqrt{D_{n}}} [∑_{s=0}^{∞} g_{s}^{+}(n)λ^{−s}/ ∑_{s=0}^{∞} ĝ_{s}^{+}(n)λ^{s}] ,

with

g_{0}^{+} = [ \begin{matrix}1 \\ 0 \end{matrix} ] , g_{1}^{+}(n) = [ −ψ̄_{−1} ; ψ̄_{n−1} ] .

Also for Q ∈ (P_{∞}^{−}/P_{0}^{−}),

g_{n}(Q) = e^{∓n log λ}\sqrt{D_{n}} [∑_{s=0}^{∞} g_{s}^{−}(n)λ^{−s}/ ∑_{s=0}^{∞} ĝ_{s}^{−}(n)λ^{s}] ,

with

g_{0}^{−} = [ \begin{matrix}0 \\ 1 \end{matrix} ] , g_{1}^{−}(n) = [ \begin{matrix}−ψ_{n} \\ ψ_{0} \end{matrix} ] .

Proof. i. Obviously,

g_{n}(Q) = C′(Q) [ \begin{matrix}T_{n}^{11}(λ) \\ T_{n}^{11}(λ)i \end{matrix} ] + C′′(Q) [ \begin{matrix}T_{n}^{12}(λ) \\ T_{n}^{22}(λ)i \end{matrix} ]

with some C′(Q) and C′′(Q). The normalisation g_{n}^{1} + g_{n}^{2}|_{n=0} = 1 implies C′′(Q) = 1 − C′(Q). Another condition T_{n}g_{0} = wg_{0} produces the system

[ \begin{matrix}T_{N}^{11} & T_{N}^{12} \\ T_{N}^{21} & T_{N}^{22} \end{matrix} ] [ \begin{matrix}C \\ 1 − C \end{matrix} ] = w [ \begin{matrix}C \\ 1 − C \end{matrix} ] .

Solving the system, we obtain the stated formulas for C(Q).

ii. The proof is identical to the proof of (i).

iii. The proof is based on explicit formulas of (i) and (ii). Using (1) of section (3), we have

T_{n,0}(λ(τ_{a}Q)) = σ_{1}\overline{T_{n,0}(λ(Q))}σ_{1}  and  w(τ_{a}Q) = \overline{w(Q)}.

From this we have 1 − \overline{C(τ_{a}Q)} = C(Q). The proof is finished by substituting T_{n}(λ(τ_{a}Q)) and C(τ_{a}Q) into explicit formula of (i). Similarly \overline{R(τ_{a}Q)} = i+R(Q). This implies the stated formula for g̃_{n}.

The last formula for g_{n}(τQ) is proved along the same lines using (2) and (6) of section 3.

(iv). It is easy to see, by perturbation arguments, that μ_{k} = λ(γ_{k}) are near the points λ_{k}^{±} = e^{i \frac{2π}{2N} k}, k = 0, …, 2N − 1 for a small potential. We will prove that |μ_{k}| = 1 always.

Let g_{n}(Q) have a pole at some point γ. Then, from the formula of (i), we have

T^{11} − T^{12} = w(Q)  and  T^{22} − T^{11} = w(Q).

Consider the special solution f_{n}(μ) = T_{n}^{(1)}(μ)−T_{n}^{(2)}(μ) of the eigenvalue problem Λ_{n}f_{n} = μf_{n}. We have

f_{0} = [ \begin{matrix}1 \\ −1 \end{matrix} ]  and  f_{N} = [ \begin{matrix}w(Q) \\ −w(Q) \end{matrix} ] .

Now the Cauchy formula (7) of section 3 to obtain

< Λf, f >_{♡}=< f, Λ^{−1}f >_{♡} +boundary terms.

Note, that Λf = μf, Λ^{−1}f = μ^{−1}f. It is easy to compute the boundary terms: to wit,

\frac{1}{N} (|w|^{2} − 1)(μ̄^{−1} − μ).

Substituting this into the Cauchy formula, we obtain

(1 − |μ|^{2}) [\frac{1}{N} (|w|^{2} − 1) − (f, f)] = 0.

This implies |μ| = 1.

(v). Consider P_{∞}^{+}/P_{0}^{+}. The relation between two asymptotic expansions follows from the formula σ_{1}g_{n}(τ_{a}Q) = g_{n}(Q). The actual form of the coefficient is derived by substituting the asymptotic expansion

g_{n}(Q) = e^{n log λ} \frac{1}{\sqrt{D_{n}}} ∑_{s=0}^{∞} g_{s}^{+}(n)λ^{−s}

into the spectral problem g_{n+1} = V (n, λ)g_{n}, where V (n, λ) = [λσ_{+} + λ^{−1}σ_{−} +Ψ_{n}]. We arrive at the reccurence relation

σ_{+}g_{s+1}^{+}(n) + σ_{−}g_{s−1}^{+}(n) + Ψ_{n}g_{s}^{+}(n) = g_{s+1}^{+}(n + 1).

The boundary condition g_{n}^{1} + g_{n}^{2}|_{n=0} = 1 implies

g_{0}^{1}(n) + g_{0}^{2}(n)|_{n=0} = 1;  g_{s}^{1}(n) + g_{s}^{2}(n)|_{n=0} = 0  for  s ≥ 1.

Starting from s = −1, reccurently, one can compute g_{0}^{+}(n), g_{1}^{+}(n), etc. □

On the curve Γ one can consider the Baker-Akhiezer (BA) function g(τ, n, t, λ) with 2N poles on the real ovals. On each oval there is just one pole. The function g(τ, n, t, λ) has the following assymptotics at the infinities:

g(τ, n, t, λ) = e^{\frac{i}{2} τ+n log λ+i(λ^{2}−1)t} \frac{1}{\sqrt{D_{n}}} ∑_{s=0}^{∞} g_{s}^{+}λ^{−s},  Q ∈ (P_{∞}^{+}),

g(τ, n, t, λ) = e^{− \frac{i}{2} τ−n log λ−i(λ^{−2}−1)t}\sqrt{D_{n}} ∑_{s=0}^{∞} g_{s}^{−}λ^{−s},  Q ∈ (P_{∞}^{−}),

g(τ, n, t, λ) = e^{− \frac{i}{2} −n log λ−i(λ^{−2}−1)t} \frac{1}{\sqrt{D_{n}}} ∑_{s=0}^{∞} ĝ_{s}^{+}λ^{s},  Q ∈ (P_{0}^{+}),

g(τ, n, t, λ) = e^{\frac{i}{2} +n log λ+i(λ^{2}−1)t}\sqrt{D_{n}} ∑_{s=0}^{∞} ĝ_{s}^{−}λ^{s},  Q ∈ (P_{0}^{−}).

The BA function with these properties exists and defined uniqly. The BA function g(τ, n, t, Q) satisfies the identities

[∂_{τ} − V_{1}]g(τ, n, t, Q) = 0,
[Δ − V_{2}]g(τ, n, t, Q) = 0,
[∂_{t} − V_{3}]g(τ, n, t, Q) = 0.

The explicit form of the V ’s is given in section 2. The Floquet solution is a particular case of BA function with the variables τ, t fixed.

6. Dual Floquet solution. Variational identity. One can write the equation V (n)g_{n} = g_{n+1} in the form*

[JΔ − JV (n)]g_{n} = 0.

Let us define the dual Floquet solution g^{+} = [g^{1+}, g^{2+}] at the point Q by

g_{n}^{+}(Q) = g_{n}(τ_{±}Q)^{T} .

Lemma 2. The dual Floquet solution g_{n}^{+}(Q) satisfies**

g_{n}^{+}(Q) [JΔ − JV (n − 1, λ)] = 0.

Proof. The equation V (n−1)g_{n−1} = g_{n} can be written as g_{n−1} = V^{−1}(n−1)g_{n}, where

V^{−1}(n − 1) = \frac{1}{\sqrt{R_{n−1}}} [ \begin{matrix}λ^{−1} & −ψ_{n−1} \\ −ψ̄_{n−1} & λ \end{matrix} ];

or, in coordinates,

g_{n−1}^{1} = \frac{1}{\sqrt{R_{n−1}}} (λ^{−1}g_{n}^{1} − ψ_{n−1}g_{n}^{2}),
g_{n−1}^{2} = \frac{1}{\sqrt{R_{n−1}}} (−ψ̄_{n−1}g_{n}^{1} + λg_{n}^{2}).

We can also rewrite it in the form

(g_{n}^{1}, g_{n}^{2}) ([ \begin{matrix}0 & Δ \\ −Δ & 0 \end{matrix} ] − \frac{1}{\sqrt{R_{n−1}}} [ \begin{matrix}ψ̄_{n−1} & λ^{−1} \\ −λ & −ψ_{n−1} \end{matrix} ]) = 0.

This is exactly the stated identity. □

We need the standard formula [KP] for variations of the quasi-momentum.

*J = iσ_{2}.

**f_{n}Δ = f_{n−1}.

Lemma 3. i. The expression g_{n}^{+}(Q)Jg_{n}(Q) does not depend on n and

g_{n}^{+}(Q)Jg_{n}(Q) =< g^{+}Jg >= Ψ(Q).

ii. The following identity holds δpΨ(Q) =< g_{n}^{+}JδV (n − 1)g_{n−1} > .

Proof. i. Can be checked using difference equation for g_{n}(Q) and g_{n}^{+}(Q).

ii. Denote by Ṽ (n, λ) and g̃_{n}(λ) deformed matrix V (n, λ) and the Floquet solution g_{n}(λ); to wit, Ṽ = V + ϵδV + o(ϵ) and g̃ = g + ϵδg + o(ϵ). Then,

g_{n}^{+}(Q) [(JΔ − JṼ (n − 1, λ))g̃_{n−1}(Q)] = 0,
[g_{n}^{+}(Q)(JΔ − JV (n − 1, λ))] g̃_{n−1}(Q) = 0.

Subtracting, we obtain

∑_{n=0}^{N−1}g_{n}^{+}(JΔg̃_{n−1}) − (g_{n}^{+}JΔ)g̃_{n−1}

= ∑_{n=0}^{N−1} g_{n}^{+} (JṼ (n − 1)g̃_{n−1}) − (g_{n}^{+}JV (n − 1)) g̃_{n−1}.

It is easy to see, that

RHS = ϵ ∑_{n=0}^{N−1} g_{n}^{+}JδV (n − 1)g_{n−1} + o(ϵ).

Using the formula

∑_{n=0}^{N−1}(g_{n}^{+}JΔg)g̃_{n−1} = ∑_{n=0}^{N−1} g_{n}^{+}(JΔg̃_{n−1}) + g_{−1}^{+}Jg̃_{−1} − g_{N−1}^{+}Jg̃_{N−1},

we obtain

LHS = g_{N−1}^{+}Jg̃_{N−1} − g_{−1}^{+}Jg̃_{−1}.

From the definition of the Floquet solution

g_{N−1}^{+} = e^{−Np}g_{−1}^{+} and g̃_{N−1} = e^{Np̃}g̃_{−1},

we have

LHS = (e^{N(p̃−p)} − 1) g_{−1}^{+}Jg̃_{−1} = ϵNδpg_{−1}^{+}Jg_{−1} + o(ϵ).

Collecting terems with ϵ, we obtain the stated identity □

Let us introduce g_{n}^{∗}(Q), the dual Floquet solution normalized by the condition < g^{∗}Jg >= 1. Obviously,

g_{n}^{∗}(Q) = \frac{g_{n}^{+}(Q)}{Ψ(Q)};

g_{n}^{∗}(Q) has poles at the branch points of the curve.

7. Hamiltonian formalism for the Ablowitz-Ladik system. As shown in [KP], the formula

ω_{0} = \frac{i}{2} ∑ res \frac{1}{Ψ(Q)} < g^{+}(n)JδV (n − 1) ∧ δg(n − 1) > \frac{dλ}{λ}.

defines a closed, nondegenerate 2 form on the space of operators Δ − V_{2} with periodic potential. Our goal to compute the residue explicitly.

Near P_{0}^{+},

(1)  \frac{1}{Ψ(Q)} = ψ_{0} + ψ_{1}λ^{1} + ⋯,  Q ∈ (P_{0}^{+}).

The identity

Ψ(τ_{±}Q) =< g^{+}(τ_{±}Q)Jg(τ_{±}Q) > =< g^{T} (Q)Jg^{+}(Q)^{T} >
= − < g^{+}(Q)Jg(Q) >= −Ψ(Q)

implies

(2)  \frac{1}{Ψ(Q)} = −ψ_{0} − ψ_{1}λ^{1} − ⋯,  Q ∈ (P_{0}^{−}).

The involution τ_{a} maps P_{∞}^{+/−} to P_{0}^{+/−}, and g_{n}(τ_{a}Q) = σ_{1}ḡ_{n}(Q). Therefore, using τ_{±}τ_{a} = τ_{a}τ_{±}, we have

Ψ(τ_{a}Q) = g^{T} (τ_{±}τ_{a}Q)Jg(τ_{a}Q) = g^{T} (τ_{a}τ_{±}Q)Jg(τ_{a}Q)
= ḡ^{T} (τ_{±}Q)σ_{1}^{T} Jσ_{1}ḡ(Q) = −\overline{Ψ(Q)}

so that, if Q ∈ (P_{∞}^{+}), then τ_{a}Q ∈ (P_{0}^{+}) and

(3)  \begin{aligned}\frac{1}{Ψ(Q)} & = −\overline{\frac{1}{Ψ(τ_{a}Q)}} = −\overline{(ψ_{0} + \frac{ψ_{1}}{λ̄^{1}} + ⋯)} \\ & = −ψ̄_{0} − \frac{ψ̄_{1}}{λ^{1}} − ⋯,  Q ∈ (P_{∞}^{+}). \end{aligned}

Again, applying τ_{±}, we have

(4)  \frac{1}{Ψ(Q)} = ψ̄_{0} + \frac{ψ̄_{1}}{λ^{1}} + ⋯,  Q ∈ (P_{∞}^{−}).

AN ADDITIONAL GIBBS’ STATE 23

It is easy to compute

ψ_{0} =< ĝ_{0}^{−T} Jĝ_{0}^{+} >= 1
ψ_{1} =< ĝ_{0}^{−T} Jĝ_{1}^{+} > + < ĝ_{1}^{−T} Jĝ_{0}^{+} >= ψ̄_{0} − ψ_{−1}, etc.

Similarly, introducing S(Q) ≡< g_{n}^{+}(Q)JδV (n − 1) ∧ δg_{n−1}(Q) >, we obtain

(5)  S(τ_{a}Q) = −\overline{S(Q)}.

One can show that, in the vicinity of the infinities:

S(Q) = c_{0} + c_{1}λ + ⋯,  Q ∈ (P_{0}^{+}),
S(Q) = d_{0} + \frac{d_{−1}}{λ} + ⋯,  Q ∈ (P_{∞}^{+});

and

S(Q) = a_{0} + a_{1}λ + ⋯  Q ∈ (P_{0}^{−}),
S(Q) = b_{0} + \frac{b_{−1}}{λ} + ⋯  Q ∈ (P_{∞}^{−}).

(5) implies d_{−k} = −c̄_{k} and b_{−k} = −ā_{k}. Computing residues, we have

res_{P_{∞}^{+}} \frac{S}{Ψ} \frac{dλ}{λ} = res_{P_{∞}^{+}} (d_{0} + ⋯) (−ψ̄_{0} − ⋯) \frac{dλ}{λ} = d_{0}ψ̄_{0},
res_{P_{∞}^{−}} \frac{S}{Ψ} \frac{dλ}{λ} = res_{P_{∞}^{−}} (b_{0} + ⋯) (ψ̄_{0} + ⋯) \frac{dλ}{λ} = −b_{0}ψ̄_{0},

and

∑_{P_{∞}^{+/−}} res \frac{S}{Ψ} \frac{dλ}{λ} = d_{0} − b_{0}.

Similarly,

res_{P_{0}^{+}} \frac{S}{Ψ} \frac{dλ}{λ} = res_{P_{0}^{+}} (c_{0} + ⋯) (ψ_{0} + ⋯) \frac{dλ}{λ} = c_{0}ψ_{0},
res_{P_{0}^{−}} \frac{S}{Ψ} \frac{dλ}{λ} = res_{P_{0}^{−}} (a_{0} + ⋯) (−ψ_{0} − ⋯) \frac{dλ}{λ} = −a_{0}ψ_{0},

and

∑_{P_{0}^{+/−}} res \frac{S}{Ψ} \frac{dλ}{λ} = c_{0} − a_{0} = −\overline{(d_{0} − b_{0})}.

24 K.L. VANINSKY

Using the formulas for Floquet solutions from Lemma 1, we have

b_{0} = < \frac{g_{0}^{+T}}{\sqrt{D_{n}}} Jδ \frac{σ_{−}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}g_{0}^{−} >

+ < \frac{g_{0}^{+T}}{\sqrt{D_{n}}} Jδ \frac{Ψ_{n−1}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}g_{1}^{−} >

+ < \frac{g_{1}^{+T}}{\sqrt{D_{n}}} Jδ \frac{σ_{+}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}g_{1}^{−} >

+ < \frac{g_{1}^{+T}}{\sqrt{D_{n}}} Jδ \frac{Ψ_{n−1}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}g_{0}^{−} >

= < \frac{1}{\sqrt{D_{n}}} δ \frac{1}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}} >

− < \frac{1}{\sqrt{D_{n}}} δ \frac{ψ̄_{n−1}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}ψ_{n−1} >

+ < \frac{ψ̄_{n−1}}{\sqrt{D_{n}}} δ \frac{1}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}}ψ_{n−1} >

− < \frac{ψ̄_{n−1}}{\sqrt{D_{n}}} δ \frac{Ψ_{n−1}}{\sqrt{R_{n−1}}} ∧ δ\sqrt{D_{n−1}} > .

After simple algebra,

∴= \frac{1}{4} < \frac{δR_{n−1}}{R_{n−1}} ∧ \frac{δD_{n−1}}{D_{n−1}} > − < \frac{1}{R_{n−1}} δψ̄_{n−1} ∧ δψ_{n−1} > .

Similarly,

d_{0} =< \sqrt{D_{n}}g_{0}^{−T} Jδ \frac{σ_{+}}{\sqrt{R_{n−1}}} ∧ δ \frac{g_{0}^{+}}{\sqrt{D_{n−1}}} >

= −\frac{1}{4} < \frac{δR_{n−1}}{R_{n−1}} ∧ \frac{δD_{n−1}}{D_{n−1}} > .

Finally,

∑_{P_{∞}^{+/−}} res \frac{S}{Ψ} \frac{dλ}{λ} = < \frac{1}{R_{n−1}} δψ̄_{n−1} ∧ δψ_{n−1} > −\frac{1}{2} < \frac{δR_{n−1}}{R_{n−1}} ∧ \frac{δD_{n−1}}{D_{n−1}} >
∑_{P_{0}^{+/−}} res \frac{S}{Ψ} \frac{dλ}{λ} = − < \frac{1}{R_{n−1}} δψ_{n−1} ∧ δψ̄_{n−1} > +\frac{1}{2} < \frac{δR_{n−1}}{R_{n−1}} ∧ \frac{δD_{n−1}}{D_{n−1}} > .

Taking the sum, we obtain

ω_{0} =< \frac{i}{R_{n−1}} δψ̄_{n−1} ∧ δψ_{n−1} > .

Lemma 5. (i) The formula

ξ_{n}(Q) = λ^{n} < g^{∗}JδV ∧ δg > \frac{dλ}{λ}

n = …, −1, 0, 1, …, defines meromorphic in Q differential form on Γ with poles at γ_{1}, ⋯, γ_{2N} and P_{∞}^{+/−}, P_{0}^{+/−}.

(ii) The symplectic 2-forms defined by the formula

ω_{n} = \frac{i}{2} ∑_{P} res ξ_{n}(Q)

can be written as

ω_{n} = −\frac{i}{2} ∑_{k=1}^{2N} λ^{n}(γ_{k})δp(γ_{k}) ∧ \frac{δλ}{λ}(γ_{k}).

Proof. (i) The poles of g_{n}^{∗} at the branch points (λ^{±}, 0) are killed by the zeros of dλ. The rest are just γ_{1}, ⋯, γ_{2N} and P_{∞}^{+/−} and P_{0}^{+/−}.

(ii) By Cauchy’s theorem,

∑_{P} res ξ_{n}(Q) + ∑_{k=1}^{2N} res ξ_{n}(Q) = 0.

Near γ_{k},

g_{n} = \frac{res g_{n}}{λ − λ(γ_{k})} + O(1).

Therefore,

δg_{n}(Q) = \frac{res g_{n}}{(λ − λ(γ_{k}))^{2}} δλ(γ_{k}) + O(1) = \frac{g_{n}}{λ − λ(γ_{k})}δλ(γ_{k}) + O(1).

Note that g_{n}^{∗}(γ_{k}) = 0 and, using Lemma 3,

res_{γ_{k}} ξ_{n}(Q) = λ^{n} < g^{∗}JδV g > ∧\frac{δλ}{λ}(γ_{k})res_{γ_{k}} \frac{dλ}{λ − λ(γ_{k})} = λ^{n}(γ_{k})δp(γ_{k}) ∧ \frac{δλ}{λ}(γ_{k}).

We are done. □

The bracket {•, •}_{ω_{0}} is constructed from the symplectic form ω_{0}

\{f, g\}_{ω_{0}} ≡ i ∑_{n=1}^{N} R_{n} (\frac{δf}{δψ_{n}} \frac{δg}{δψ̄_{n}} − \frac{δf}{δψ̄_{n}} \frac{δg}{δψ_{n}} ) .

The original AL flow from section 2 can be written

ψ^{•}_{n} = \{ψ_{n}, H\}_{ω_{0}}, H = N (I_{2} + \overline{I}_{2} − 2I_{0} − 2\overline{I}_{0}) .

The phase flow is also Hamiltonian

ψ^{•}_{n} = \{ψ_{n}, P \}_{ω_{0}}, P = N(I_{0} + \overline{I}_{0}).

8. Embedding of the AL system into the function space. First, we introduce interpolating trigonometrical polynomials.

Let f(x) be a smooth 1-periodic complex function and associate to it the sequence of interpolating trigonometrical polynomials

f_{N}(x) = ∑_{|k|≤m} e^{2πik}f̂_{N}(k), N = 2m + 1, m = 1, 2 ….

These have the property that f_{N}(x) = f(x) for x ∈ 𝕊_{N} = \{x ∈ 𝕊 : x = \frac{n}{N} , n = 0, …, N − 1}. Let M_{N} be a space of such trigonometric polynomials of degree m; it is in one-to-one correspondence with the space M_{N} of N-periodic complex sequences:

f_{N}(x) ⟷ f_{n}, f_{N}(x) ∈ M_{N}, f_{n} ∈ M_{N}

if we put ϵf_{N}(\frac{n}{N}) = f_{n}, n = 0, …, N − 1.

Introduce H_{1}^{N}(ψ) ≡ H_{1}^{N}(ϵψ). The region

B^{N} = {ψ_{N} ∈ M_{N} : NH_{1}^{N}(tψ) < ∞, for all 0 ≤ t ≤ 1}

is called the ”box” of the space M_{N}; the trigonometric polynomial ψ_{N} belongs B^{N} if and only if |ψ_{N}(x)| < N, x ∈ 𝕊_{N}. The map ψ_{N}(x) ↔ ψ_{n} allows us to define the flow e^{tX_{3}^{N}} on B^{N} by the formula

e^{tX_{3}^{N}} ψ_{N} ≡ ϵ^{−1}e^{θX_{3}^{N}} ϵψ_{N}, θ = ϵ^{−2}t.

Evidently B^{N} is invariant under the flow ^{tX_{3}^{N}}. Due to the natural embedding M_{N} ⊂ M, the dynamics e^{tX_{3}^{N}} also can be defined by the formula e^{tX_{3}^{N}} ψ ≡ e^{tX_{3}^{N}} ψ_{N} for any function ψ ∈ M which satisfies the inequality |ψ(x)| < N, x ∈ 𝕊_{N}.

We define on M_{N} the volume form

d vol^{N} ≡ \frac{1}{N!D_{N}(ψ)} ⋀_{x∈𝕊_{N}}iδψ(x) ∧ δψ̄(x),

where D_{N}(ψ) ≡ D_{N}(ϵψ) and the functional H_{5}^{N}(ψ) is by definition H_{5}^{N}(ϵψ). The flow e^{θX_{3}} preserves both H_{3}(ψ_{n}, ψ̄_{n}) and the volume form

d vol = \frac{1}{N!D_{N}} ⋀_{n} iδψ_{n} ∧ δψ̄_{n}.

Therefore, the volume form d vol^{N} = ϵ^{−2N}d vol and the functional H_{5}^{N} are invariant under the flow e^{tX_{3}^{N}}. The finite measure dμ^{N}(ψ, ψ̄) on B^{N} with density

e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)}d vol^{N} = e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)} \frac{1}{N!D_{N}(ψ)} ⋀_{x∈𝕊_{N}}iδψ(x) ∧ δψ̄(x)

is also invariant under the flow e^{tX_{3}^{N}}.

To ensure proper analytic control* we need to introduce the function h(x) (fig. 5)

fig. 5

and we define the invariant functional χ_{K}^{N}(ψ) ≡ h(NH_{1}^{N})h(−N^{3}H_{3}^{N}). On the box B^{N} the probability measure dμ_{K}^{N}(ψ, ψ̄)

\frac{1}{Ξ_{K}^{N}} χ_{K}^{N}(ψ)e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)}d vol^{N},

where Ξ_{K}^{N} is a normalisating factor**.

Due to the embedding M_{N} ⊂ M, the measure dμ_{K}^{N}(ψ, ψ̄) can be transfered to the whole of M. We define on M the probability measure dμ_{K}(ψ, ψ̄) with the density

\frac{1}{Ξ_{K}} χ_{K}(ψ)e^{− \frac{1}{2} H_{5}}d vol = \frac{1}{Ξ_{K}} χ_{K}(ψ)e^{− \frac{1}{2} H_{5}} \frac{1}{∞!} ⋀_{x∈𝕊} iδψ(x) ∧ δψ̄(x),

where χ_{K}(ψ) = h(H_{1})h(H_{3}) and Ξ_{K} is a normalisating factor.

*Similar cut-off was introduced in [MCV2].

**The precise definition of Ξ_{K}^{N} and Ξ_{K} will be given in Lemma 8.

In order to prove invariance of the measure dμ_{K}, we use the method of weak solutions introduced by McKean, [MC]. The measure dμ_{K}^{N} on the initial data defines the measure dM_{K}^{N} in the space of paths C ([0, T] → M) for T > 0. All information about the measure dμ_{K}^{N} and the flow e^{tX_{3}^{N}} is encoded now into the measure dM_{K}^{N}. The proof of invariance takes three steps. Step 1. The measures dμ_{K}^{N} converge to dμ_{K} weakly in H^{s}, 1 ≤ s < \frac{3}{2}, as N → ∞. Step 2. For any fixed K, the family dM_{K}^{N}, N = 1, 2 … is tight and converges to measure dM_{K}. The stationary measure dM_{K} for fixed t has marginal distribution dμ_{K}. Step 3. The measure dM_{K} is supported on the solutions of the NSL flow.

All three steps will be completed in subsequent sections. From the invariance of the measure dμ_{K}, it is easy to infer the invariance of the the desired Gibbs’ state with the density \frac{1}{Ξ}e^{− \frac{1}{2} H_{5}}d vol.

9. Convergence of AL Gibbs’ state to the NLS Gibbs’ state. In the previous section, we introduced the family of probability measures dμ_{K}^{N}(ψ, ψ̄) on B^{N} ⊂ M with the densities*

χ_{K}^{N}(ψ)e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)}d vol^{N} = χ_{K}^{N}(ψ)e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)} \frac{1}{N! D_{N}(ψ)} ⋀_{x∈𝕊_{N}}iδψ(x) ∧ δψ̄(x).

We will show that dμ_{K}^{N} → dμ_{K}, weakly in H^{s}, 1 ≤ s < \frac{3}{2}, as N → ∞, where dμ_{K}(ψ, ψ̄) has the density

χ_{K}(ψ)e^{− \frac{1}{2} H_{5}(ψ)}d vol = χ_{K}(ψ)e^{− \frac{1}{2} H_{5}(ψ)} \frac{1}{∞!} ⋀_{x∈𝕊}iδψ(x) ∧ δψ̄(x).

The cut-off in K can be removed easily and dμ_{K} converges to dμ.

We split the integral H_{5}^{N} into a quadratic part G^{N}(ψ) and a nonlinear part R^{N}(ψ). Using the explicit expression for the integrals I’s, we have

H_{5}^{N}(ψ_{n}) = G^{N}(ψ_{n}) + R^{N}(ψ_{n})

= ∑(ψ_{n+1}ψ̄_{n−1} + ψ̄_{n+1}ψ_{n−1}) − 4(ψ_{n}ψ̄_{n+1} + ψ̄_{n}ψ_{n+1}) + 6|ψ_{n}|^{2}

− ∑(ψ_{n+1}ψ̄_{n−1} + ψ̄_{n+1}ψ_{n−1})|ψ_{n}|^{2} − \frac{1}{2} ∑(ψ_{n}ψ̄_{n−1})^{2} + (ψ̄_{n}ψ_{n−1})^{2}

− ∑ 6|ψ_{n}|^{2} + 12NI_{0}.

It is easy to check that

G^{N}(ψ_{n}) = ∑_{n=0}^{N−1} |ψ_{n+1} − 2ψ_{n} + ψ_{n−1}|^{2}.

*To simplify notation, we omit the normalization factors Ξ for a moment.

AN ADDITIONAL GIBBS’ STATE 29

Introduce

I^{N}(ψ_{n}) = ∑_{n=0}^{N−1} |ψ_{n}|^{2}

and also

G^{N}(ψ) = G^{N}(ψ_{n}), R^{N}(ψ) = R^{N}(ψ_{n}), I^{N}(ψ) = I^{N}(ψ_{n}),

where ψ_{n} = ϵψ(\frac{n}{N}). With the new notations,

χ_{K}^{N}(ψ)e^{− \frac{N^{5}}{2} H_{5}^{N} (ψ)}d vol^{N}
=χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2}[−NI^{N} (ψ)+N^{5}R^{N} (ψ)]}

× e^{− \frac{1}{2}[N^{5}G^{N} (ψ)+NI^{N} (ψ)]} \frac{1}{N!} ⋀_{x∈𝕊_{N}}iδψ(x) ∧ δψ̄(x)

≍χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N} (ψ)} × dγ^{N}(ψ, ψ̄).

Now we are ready to prove Lemma 6. The family of Gaussian probability measures d γ^{N}(ψ, ψ̄) with the densities

e^{− \frac{1}{2}[N^{5}G^{N} (ψ)+NI^{N} (ψ)]} \frac{1}{N!} ⋀_{x∈𝕊_{N}}iδψ(x) ∧ δψ̄(x)

converges weakly in H^{s}, s < \frac{3}{2}, as N → ∞, to the probability measure dγ(ψ, ψ̄):

e^{− \frac{1}{2} ∫ |ψ′′|^{2}+|ψ|^{2}} \frac{1}{∞!} ⋀_{x∈𝕊}iδψ(x) ∧ δψ̄(x).

Proof. For a smooth ψ(x) = ∑_{k} e^{2πixk}ψ̂(k) and N → ∞,

N^{5}G^{N}(ψ) + NI^{N}(ψ)

= N^{5}ϵ^{2} ∑ |ψ (n + 1/N) − 2ψ (n/N) + ψ (n − 1/N) |^{2} + Nϵ^{2} ∑ |ψ (n/N) |^{2}

≈ N^{3}ϵ^{4} ∑ |ψ′′ (n/N) |^{2} + ϵ ∑ |ψ (n/N) |^{2} → ∫_{0}^{1} |ψ′′(x)|^{2} + |ψ(x)|^{2}d x.

This explains on a formal level, the convergence d γ^{N} → d γ.

To obtain the full proof, we write all measures in terms of Fourier coefficients. First we note that

∫_{0}^{1} |ψ′′(x)|^{2} + |ψ(x)|^{2}dx = ∑_{k}(16π^{4}k^{4} + 1)|ψ̂(k)|^{2} = ∑_{k} σ^{2}(k)|ψ̂(k)|^{2}.

Using the identity

\frac{1}{N} ∑_{p=0}^{N−1} |ψ(n/N)|^{2} = ∑_{|k|≤m} |ψ̂_{N}(k)|^{2},

for ψ ∈ H^{1}, we have

N^{5}G^{N}(ψ) + NI^{N}(ψ) = ∑_{|k|≤m} [N^{4}|e^{2πik/N} − 2 + e^{−2πik/N}|^{2} + 1] |ψ̂_{N}(k)|^{2}

= ∑_{|k|≤m} σ_{N}^{2}(k)|ψ̂_{N}(k)|^{2}.

Therefore, the measures d γ^{N}(ψ, ψ̄) can be written as*

≍ e^{− \frac{1}{2} ∑_{|k|≤m} σ_{N}^{2} (k)|ψ̂(k)|^{2}} ⋀_{|k|≤m}iδψ̂(k) ∧ δ ¯ψ̂(k).

Similarly, for dγ(ψ, ψ̄)

≍ e^{− \frac{1}{2} ∑_{k} σ^{2}(k)|ψ̂(k)|^{2}} ⋀_{k} iδψ̂(k) ∧ δ ¯ψ̂(k).

For any fixed k,

σ_{N}^{2}(k) = N^{4}|e^{2πik/N} − 2 + e^{−2πik/N}|^{2} + 1 ⟶ 16π^{4}k^{4} + 1 = σ^{2}(k),

as N → ∞, whence the convergence of measures on finite-dimentional subspaces generated by the Fourier harmonics.

To prove tightness of the measures γ^{N}, we introduce a ”brick” B_{a} ≡ {ψ : |ψ̂(k)| ≤ a_{k}, k ∈ ℤ}, where a_{k} ≥ 0. A brick is compact in H^{s} if and only if

∑_{k}(1 + k^{2})^{s}a_{k}^{2} < ∞.

*The sign ≍ means up to unessential constant real factor.

Let us estimate

γ^{N}(B_{a}) = ∏_{|k|≤m} \frac{σ_{N}^{2}}{2π} ∫_{|ψ̂(k)|^{2}≤a_{k}^{2}} e^{− \frac{σ^{2}_{N}}{2} |ψ̂(k)|^{2}} \frac{i}{2} δψ̂(k) ∧ δ ¯ψ̂(k)

= ∏_{|k|≤m} [1 − e^{−^{\frac{σ^{2}_{N} (k)a^{2}_{k}}{2}}} ] .

The inequalities (1 − b_{1}) × … × (1 − b_{n}) ≥ 1 − b_{1} − … − b_{n}, b_{k} ≥ 0, and e^{−x} ≤ p!x^{−p}, x ≥ 0, imply

γ^{N}(B_{a}) ≥1 − ∑_{|k|≤m} e^{−^{\frac{σ^{2}_{N} (k)a^{2}_{k}}{2}}}

≥1 − 2^{p}p! ∑_{|k|≤m} \frac{1}{σ_{N}^{2p}(k)a_{k}^{2p}} .

Now we obtain an estimate for σ_{N}^{2}(k) = 4N^{4}(cos \frac{2πk}{N} − 1)^{2} + 1, |k| ≤ m. Obviously,

a′x^{4} ≤ (cos x − 1)^{2} ≤ b′x^{4} for − π ≤ x ≤ π.

Therefore,

ak^{4} + 1 ≤ σ_{N}^{2}(k) ≤ bk^{4} + 1.

This, together with

a_{k}^{2} = \frac{C}{(|k| + 1)^{α}} , C > 0,

implies

γ^{N}(B_{a}) ≥ 1 − \frac{2^{p}p!}{C^{p}} ∑_{k} \frac{(|k| + 1)^{αp}}{(ak^{4} + 1)^{p}} .

Pick any s < \frac{3}{2}. Then 2s + 1 < 4. Pick any α such that 2s + 1 < α < 4. Then the bricks B_{a} are compact in H^{s}. Now pick p such that (4 − α)p > 1. Then the sum in the last estimate converges. Chose C so large as to make γ^{N}(B_{a}) arbitrary close to 1. □

Lemma 7. (i). For any ψ ∈ H^{1} the functionals

χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2}[−NI^{N} (ψ)+N^{5}R^{N} (ψ)]} = χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N} (ψ)}

converge to

χ_{K}(ψ)e^{− \frac{1}{2} ∫ −|ψ|^{2}+2|ψ|^{6}+6|ψ′|^{2}|ψ|^{2}+(ψ̄′ψ+ψ′ψ̄)^{2}} = χ_{K}(ψ)e^{− \frac{1}{2} W(ψ)} < \sqrt{e},

as N → ∞.

(ii).

1 ≥ D_{N}(ψ) ≥ e^{− \frac{K}{N}}

and

W^{N}(ψ) ≥ −c_{4}(K),

uniformly for all N, provided NH_{1}^{N} ≤ K, −N^{3}H_{3}^{N} ≤ K.

Proof. (i). The nonlinear term

R^{N}(ψ_{n}, ψ̄_{n}) = − ∑(ψ_{n+1}ψ̄_{n−1} + ψ̄_{n+1}ψ_{n−1})|ψ_{n}|^{2}

− \frac{1}{2} ∑(ψ_{n}ψ̄_{n−1})^{2} + (ψ̄_{n}ψ_{n−1})^{2}

− 6 ∑ |ψ_{n}|^{2} + 12NI_{0}

can be expressed as

R^{N}(ψ_{n}, ψ̄_{n}) =

(A)  + ∑ [(ψ_{n+1} − ψ_{n})(ψ̄_{n} − ψ̄_{n−1}) + (ψ̄_{n+1} − ψ̄_{n})(ψ_{n} − ψ_{n−1})] |ψ_{n}|^{2}
(B)  − ∑ [(ψ_{n+1} + ψ_{n−1} − 2ψ_{n})ψ̄_{n} + (ψ̄_{n+1} + ψ̄_{n−1} − 2ψ̄_{n})ψ_{n}] |ψ_{n}|^{2}
(C)  −\frac{1}{2} ∑ ψ_{n}^{2}(ψ̄_{n−1} − ψ̄_{n})^{2} + ψ̄_{n}^{2}(ψ_{n−1} − ψ_{n})^{2}
(D)  − ∑ ψ_{n}^{2}ψ̄_{n}(ψ̄_{n−1} − ψ̄_{n}) + ψ̄_{n}^{2}ψ_{n}(ψ_{n−1} − ψ_{n})
(E)  −3 ∑ |ψ_{n}|^{4} − 6 ∑ |ψ_{n}|^{2} + 12NI_{0}.

As in the continuous case, if one counts the difference of ψ’s in two neighboring points (ψ_{n+1} − ψ_{n}) and the function ψ_{n} itself of weight 1, then the terms A, B and C are isobaric polynomials of degree 6. The term D is isobaric polynomial of degree 5.

The term A can be reduced to the form

A(ψ_{n}, ψ̄_{n}) = ∑ |ψ_{n+1} − ψ_{n−1}|^{2}|ψ_{n}|^{2} − ∑{|ψ_{n+1} − ψ_{n}|^{2} + |ψ_{n−1} − ψ_{n}|^{2}}|ψ_{n}|^{2}.

We transform the terms B and D to a more convinient form also:

B(ψ_{n}, ψ̄_{n}) = ∑(ψ_{n+1} − ψ_{n})^{2}ψ̄_{n}^{2} + (ψ̄_{n+1} − ψ̄_{n})^{2}ψ_{n}^{2}
+ ∑ |ψ_{n+1} − ψ_{n}|^{2} [ψ_{n+1}(ψ̄_{n} + ψ̄_{n+1}) + ψ̄_{n+1}(ψ_{n} + ψ_{n+1})]
= ∑ [(ψ_{n+1} − ψ_{n})ψ̄_{n} + (ψ̄_{n+1} − ψ̄_{n})ψ_{n}]^{2}
+2 ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n+1}|^{2}
+ ∑ |ψ_{n+1} − ψ_{n}|^{2} [(ψ_{n+1} − ψ_{n})ψ̄_{n} + (ψ̄_{n+1} − ψ̄_{n})ψ_{n}]

and

D(ψ_{n}, ψ̄_{n}) = \frac{1}{2} ∑ ψ_{n}^{2}(ψ̄_{n} − ψ̄_{n−1})^{2} + ψ̄_{n}^{2}(ψ_{n} − ψ_{n−1})^{2}
+2 ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n}|^{2}
+\frac{1}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2} [ψ̄_{n}(ψ_{n+1} − ψ_{n}) + ψ_{n}(ψ̄_{n+1} − ψ̄_{n})]
−\frac{1}{2} ∑ |ψ_{n}|^{2} [|ψ_{n+1} − ψ_{n}|^{2} − |ψ_{n−1} − ψ_{n}|^{2}] .

The first term in the formula is C with the opposite sign. Finally,

R(ψ_{n}, ψ̄_{n}) =
(1)  −3 ∑ |ψ_{n}|^{4} − 6 ∑ |ψ_{n}|^{2} + 12NI_{0}
(2)  + ∑ |ψ_{n+1} − ψ_{n−1}|^{2}|ψ_{n}|^{2}
(3)  +\frac{3}{2} ∑ |ψ_{n−1} − ψ_{n}|^{2}|ψ_{n}|^{2}
(4)  +\frac{1}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n}|^{2}
(5)  + ∑ [(ψ_{n+1} − ψ_{n})ψ̄_{n} + (ψ̄_{n+1} − ψ̄_{n})ψ_{n}]^{2}
(6)  +\frac{3}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2} [ψ̄_{n}(ψ_{n+1} − ψ_{n}) + ψ_{n}(ψ̄_{n+1} − ψ̄_{n})]

The advantage of such a representation is that the terms (1)-(5) are nonnegative. The term (6) is the only term which is not sign-definite. This term vanishes when N → ∞. Finally, as N → ∞

N^{5}R^{N}(ψ) ⟶ ∫_{0}^{1} 2|ψ|^{6} + 6|ψ′|^{2}|ψ|^{2} + (ψ̄′ψ + ψ′ψ̄)^{2} ,

NI^{N}(ψ) ⟶ ∫_{0}^{1} −|ψ|^{2}

D_{N}(ψ) ⟶ 1.

(ii). From the definition H_{1} = N(I_{0} + Ī_{0}) = − log D we have H_{1}^{N}(ψ) = − log D_{N}(ψ). The inequality NH_{1}^{N} ≤ K implies, that 1 ≥ D_{N} ≥ e^{− \frac{K}{N}}.

To obtain the lower bound for R^{N}(ψ_{n}, ψ̄_{n}), we prove first that

(7)  −3 ∑ |ψ_{n}|^{4} − 6 ∑ |ψ_{n}|^{2} + 12NI_{0} ≥ 2 ∑ |ψ_{n}|^{6}.

34 K.L. VANINSKY

Indeed, for 0 < x < 1

− log(1 − x) ≥ x + \frac{x^{2}}{2} + \frac{x^{3}}{3} .

Then,

−6 log(1 − |ψ_{n}|^{2}) ≥ 6|ψ_{n}|^{2} + 3|ψ_{n}|^{4} + 2|ψ_{n}|^{6}.

This and 12NI_{0} = −6 log ∏(1 − |ψ_{n}|^{2}) imply the inequality (7). Therefore, the term (1) is nonegative.

We have just one term (6) which is not sign-definite

\frac{3}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2} [ψ_{n}ψ̄_{n+1} + ψ̄_{n}ψ_{n+1} − 2|ψ_{n}|^{2}] .

Using the identity

|ψ_{n+1} − ψ_{n}|^{2} − |ψ_{n+1}|^{2} − |ψ_{n}|^{2} = ψ_{n+1}ψ̄_{n} + ψ̄_{n+1}ψ_{n}

and adding to it the two positive terms (3) and (4), we have

\frac{3}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2} (ψ_{n}ψ̄_{n+1} + ψ̄_{n}ψ_{n+1} − 2|ψ_{n}|^{2})
+\frac{3}{2} ∑ |ψ_{n−1} − ψ_{n}|^{2}|ψ_{n}|^{2}
+\frac{1}{2} ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n}|^{2}
≥ − 4 ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n}|^{2}.

Therefore,

R(ψ_{n}, ψ̄_{n}) ≥ −4 ∑ |ψ_{n+1} − ψ_{n}|^{2}|ψ_{n}|^{2}.

and

N^{5}R(ψ) ≥ −4 ∑ N|ψ(n + 1/N) − ψ(n/N)|^{2}|ψ(n/N)|^{2}.

Now we use the constraint −NH_{3}^{N} ≤ K and the explicit formula for H_{3}:

−H_{3}^{N}(ψ_{n}, ψ̄_{n}) = ∑ |ψ_{n+1} − ψ_{n}|^{2} − 2|ψ_{n}|^{2} − 2 log(1 − |ψ_{n}|^{2}).

The estimate − log(1 − x) ≥ x + \frac{x^{2}}{2} for 0 ≤ x < 1 implies −2 log(1 − |ψ_{n}|^{2}) ≥ 2|ψ_{n}|^{2} + |ψ_{n}|^{4}.

AN ADDITIONAL GIBBS’ STATE 35

Therefore,

−H_{3}^{N}(ψ_{n}, ψ̄_{n}) ≥ ∑ |ψ_{n+1} − ψ_{n}|^{2} + |ψ_{n}|^{4}.

Finally,

K ≥ −N^{3}H_{3}^{N}(ψ) ≥ ∑ N|ψ(n + 1/N) − ψ(n/N)|^{2} + N^{−1}|ψ(n/N)|^{4}. This inequality also provides the estimate for max |ψ(n/N)|*. Indeed,

\frac{1}{N} ∑ |ψ(n/N)|^{4} ≤ K,

implies that there exists n′′ such that |ψ(n′′/N)| ≤ K^{1/4}. Then, for any n′,

ψ(n′/N) = ∑_{n′≤n<n′′} [ψ(n + 1/N) − ψ(n/N)] + ψ(n′′/N)

and, by Schwartz’s inquality,

|ψ(n′/N)| ≤ ∑ |ψ(n + 1/N) − ψ(n/N)|\begin{matrix}√ & \overline{N} \\ √ & \overline{N} \end{matrix} + |ψ(n′′/N)|
≤ \sqrt{∑ N|ψ(n + 1/N) − ψ(n/N)|^{2}} + |ψ(n′′/N)|
≤ c_{2}(K).

Therefore,

N^{5}R^{N}(ψ) ≥ −4 max |ψ(n/N)|^{2} ∑ N|ψ(n + 1/N) − ψ(n/N)|^{2}
≥ −4c_{2}^{2}(K)K = −c_{3}(K).

Also,

NI^{N}(ψ) = ∑ N^{−1}|ψ(n/N)|^{2} ≤ c_{2}^{2}(K).

Finally,

W^{N}(ψ) = −NI^{N}(ψ) + N^{5}R^{N}(ψ) ≥ −c_{2}^{2}(K) − c_{3}(K) = −c_{4}(K).

□ Lemma 8. (i) The probability measures

*This is similar to the continuum case when the H^{1}-norm provides a bound in the sup-norm.

dμ_{K}^{N} = \frac{1}{Ξ_{K}^{N}} χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N} (ψ)}dγ^{N}(ψ, ψ̄),

with

Ξ_{K}^{N} ≡ ∫_{M} χ_{K}^{N}(ψ)\frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N} (ψ)}dγ^{N}(ψ, ψ̄)

converge weakly in H^{s}, 1 ≤ s < \frac{3}{2} as N → ∞ to the probability measure

dμ_{K} = \frac{1}{Ξ_{K}} χ_{K}(ψ)e^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄),

with

Ξ_{K} ≡ ∫_{M} χ_{K}(ψ)e^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄).

(ii). The measure dμ_{K} converges in H^{s}, 1 ≤ s < \frac{3}{2} as K → ∞ to the probability measure

dμ = \frac{1}{Ξ}e^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄),

with

Ξ ≡ ∫_{M} e^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄).

Proof. (i). Take M = H^{s} × H^{s}, 1 ≤ s ≤ \frac{3}{2} and f bounded and continuous. We will prove that

(8)  ∫_{M} fχ_{K}^{N} \frac{1}{D_{N}(ψ)} e^{− \frac{1}{2} W^{N} (ψ)}dγ^{N}(ψ, ψ̄) ⟶ ∫_{M} fχ_{K}e^{− \frac{1}{2} W}dγ(ψ, ψ̄),

as N → ∞. This implies that Ξ_{K}^{N} → Ξ_{K} as N → ∞. Then, from (8), we have

∫ f dμ_{K}^{N} =\frac{1}{Ξ_{K}^{N}} ∫ f χ_{K}^{N} \frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N}} dγ^{N}(ψ, ψ̄)

⟶ \frac{1}{Ξ_{K}} ∫ fχ_{K}e^{− \frac{1}{2} W}dγ(ψ, ψ̄) × \frac{Ξ_{K}}{Ξ_{K}^{N}} = ∫ fdμ_{K} × \frac{Ξ_{K}}{Ξ_{K}^{N}} ,

as N → ∞ and weak convergence is proved.

To begin with,

∫ fχ_{K}^{N} \frac{1}{D_{N}(ψ)}e^{− \frac{1}{2} W^{N} (ψ)}dγ^{N}(ψ, ψ̄)

= ∫ f (\frac{1}{D_{N}(ψ)} − 1) χ_{K}^{N}e^{− \frac{1}{2} W^{N}} dγ^{N} + ∫ fχ_{K}^{N}e^{− \frac{1}{2} W^{N}} dγ^{N}(ψ, ψ̄).

The first term can be estimated using the inequality of Lemma 8, item (ii):

∴ ≤ ∥f∥_{∞} e^{\frac{1}{2} c_{4}(K)} ∫_{M} |\frac{1}{D_{N}(ψ)} − 1| χ_{K}^{N}dγ^{N}(ψ, ψ̄) = o(1),

as N → ∞. As to the second term,

∫ fχ_{K}^{N}e^{− \frac{1}{2} W^{N}} dγ^{N}(ψ, ψ̄) = ∫ fχ_{K}e^{− \frac{1}{2} W}dγ^{N}(ψ, ψ̄)
(9)  + ∫ fχ_{K}^{N} (e^{− \frac{1}{2} W^{N}} − e^{− \frac{1}{2} W}) dγ^{N}(ψ, ψ̄)
(10)  + ∫ f (χ_{K}^{N} − χ_{K}) e^{− \frac{1}{2} W}dγ^{N}(ψ, ψ̄).

The first integral on the right converges to

∫ fχ_{K}e^{− \frac{1}{2} W}dγ(ψ, ψ̄).

The other two integrals vanish as N → ∞. Indeed, (9) can be overestimated by

(11)  \frac{∥f∥_{∞}}{2} e^{\frac{c_{4}(K)}{2}} ∫ χ_{K}^{N}|W^{N} − W|d γ^{N}(ψ, ψ̄).

Using the explicit expressions for W^{N} and W, we have

W^{N} − W =
(12)  − ∑ N^{−1}|ψ(n/N)|^{2} + ∫ |ψ|^{2}dx
(13)  −3 ∑ N|ψ(n/N)|^{4} − 6 ∑ N^{3}|ψ(n/N)|^{2} + 12N^{6}I_{0}(ψ) − 2 ∫ |ψ|^{6}dx
(14)  + ∑ N|ψ(n + 1/N) − ψ(n − 1/N)|^{2}|ψ(n/N)|^{2} − 4 ∫ |ψ′|^{2}|ψ|^{2}dx
(15) +\frac{3}{2} ∑ N|ψ(n − 1/N) − ψ(n/N)|^{2}|ψ(n/N)|^{2} − \frac{3}{2} ∫ |ψ′|^{2}|ψ|^{2}dx
(16) +\frac{1}{2} ∑ N|ψ(n + 1/N) − ψ(n/N)|^{2}|ψ(n/N)|^{2} − \frac{1}{2} ∫ |ψ′|^{2}|ψ|^{2}dx
(17)  + ∑ N[(ψ(n + 1/N) − ψ(n/N))ψ̄(n/N)+
+ (ψ̄(n + 1/N) − ψ̄(n/N))ψ(n/N)]^{2} − ∫ [ψ′ψ̄ + ψ̄ψ′]^{2}dx
(18) +\frac{3}{2} ∑ N|ψ(n + 1/N) − ψ(n/N)|^{2}[ψ̄(n/N)(ψ(n + 1/N) − ψ(n/N))+
+ ψ(n/N)(ψ̄(n + 1/N) − ψ̄(n/N))].

Here, each term contributes nothing when N → ∞; for example, substituting (12) into (11):

≤ ∫_{M} |∑ N^{−1}|ψ(n/N)|^{2} − ∑ ∫_{n/N}^{n+1/N} |ψ|^{2}dx| dγ^{N}(ψ, ψ̄)
≤ ∑ ∫_{M} |N^{−1}|ψ(n/N)|^{2} − ∫_{n/N}^{n+1/N} |ψ|^{2}dx| dγ^{N}(ψ, ψ̄)
≤N ∫_{M} |∫_{0}^{1/N} |ψ(x)|^{2} − |ψ(0)|^{2}dx| dγ^{N}(ψ, ψ̄)
≤N ∫_{0}^{1/N}∫_{M} ||ψ(x)|^{2} − |ψ(0)|^{2}| dγ^{N}(ψ, ψ̄)dx.

AN ADDITIONAL GIBBS’ STATE 39

Now, by Schwartz, for 0 ≤ |x| ≤ 1/N, we have

(∫_{M} ||ψ(x)|^{2} − |ψ(0)|^{2}| dγ^{N})^{2} ≤ ∫_{M} (|ψ(x)|^{2} − |ψ(0)|^{2})^{2} dγ^{N}
= ∫_{M} |ψ(x)|^{4} − 2|ψ(x)|^{2}|ψ(0)|^{2} + |ψ(0)|^{4}dγ^{N}
=2 ∫_{M} |ψ(0)|^{4} − |ψ(x)|^{2}|ψ(0)|^{2}dγ^{N}

The stationary Gaussian measure dγ^{N} has the spectral representation

ψ_{N}(x) = ∑_{|k|≤m} e^{2πikx}ψ̂(k)

where ψ̂ are complex isotropic Gaussian variables E^{N}|ψ̂(k)|^{2} = σ_{N}^{−2}(k). Now

|ψ(0)|^{2}|ψ(x)|^{2} = ∑_{\begin{matrix}k_{1},k_{2} \\ k_{3},k_{4} \end{matrix}} e^{2πix(k_{1}−k_{2})}ψ̂(k_{1})\overline{ψ̂}(k_{2})ψ̂(k_{3})\overline{ψ̂}(k_{4}).

The terms in the last sum contributing into the expectation E^{N} are

k_{1} = k_{2} k_{1} = k_{4} k_{1} = k_{2} = k_{3} = k_{4}.
k_{3} = k_{4}  k_{3} = k_{2}
k_{1} ≠ k_{3}  k_{1} ≠ k_{3}

Now

E^{N}|ψ(0)|^{2}|ψ(x)|^{2} = ∑_{k_{1}≠k_{3}} E^{N}|ψ̂(k_{1})|^{2}E^{N}|ψ̂(k_{3})|^{2}

+ ∑_{k_{1}≠k_{2}} e^{2πix(k_{1}−k_{2})}E^{N}|ψ̂(k_{1})|^{2}E^{N}|ψ̂(k_{2})|^{2} + ∑_{k_{1}} E^{N}|ψ̂(k_{1})|^{4},

So,

E^{N}|ψ(0)|^{4} − E^{N}|ψ(x)|^{2}|ψ(0)|^{2}

= ∑_{k_{1}≠k_{2}} [1 − e^{2πix(k_{1}−k_{2})}] E^{N}|ψ̂(k_{1})|^{2}E^{N}|ψ̂(k_{2})|^{2}

= ∑_{|k_{i}|≤m} |1 − e^{2πix(k_{1}−k_{2})}|σ_{N}^{−2}(k_{1})σ_{N}^{−2}(k_{2}).

Using the estimate ak^{4} + 1 ≤ σ_{N}^{2}(k) of Lemma 7, uniformly in 0 ≤ x ≤ 1/N, the last sum can be overestimated as

∴ ≤ ∑_{k_{i}∈ ℤ^{1}} |1 − e^{2πix(k_{1}−k_{2})}| \frac{1}{ak_{1}^{4} + 1} \frac{1}{ak_{2}^{4} + 1} = o(1),

as N → ∞. This implies that the term (12) contributes nothing when N → ∞. To estimate the contribution of (13), expand the log in I_{0}:

12N^{6}I_{0}(ψ) = − 6N^{5} log ∏(1 − N^{−2}|ψ(n/N)|^{2})
=6N^{5}[N^{−2}|ψ(n/N)|^{2} + \frac{N^{−4}}{2} |ψ(n/N)|^{4} + \frac{N^{−6}}{3} |ψ(n/N)|^{6}
+\frac{f^{(iv)}}{4!} N^{−8}|ψ(n/N)|^{8}]

where f(x) = − log(1 − x) and the fourth derivative f^{(iv)} = 3!(1 − x)^{−4} in the remainder term is bounded since |ψ(n/N)| ≤ c_{2}(K) and N^{−2}|ψ(n/N)|^{2} ≤ θ < 1.

Now for (13), we have

2 ∑ N^{−1}|ψ(n/N)|^{6} − 2 ∫_{0}^{1} |ψ|^{6}dx + \frac{6}{4!}N^{−3} ∑ f^{(iv)}|ψ(n/N)|^{6}.

The part with the the fourth derivative can be overestimated by

∴ ≤ \frac{6}{4!}N^{−3} \frac{3!}{(1 − θ)^{4}} c_{2}^{6}(K)N = o(1),

as N → ∞. The contribution to (11) of first two terms can be overestimated by

∴ ≤ 2 |∫_{M} ∑ N^{−1}|ψ(n/N)|^{6} − ∫_{n/N}^{n+1/N} |ψ|^{6}dx| dγ^{N}

≤ 2N ∫_{0}^{1/N} ∫_{M} ||ψ(x)|^{6} − |ψ(0)|^{6}| dγ^{N}dx.

Again, by Schwarz,

(∫_{M} ||ψ(x)|^{6} − |ψ(0)|^{6}| dγ^{N})^{2}
≤ 2 ∫_{M} |ψ(0)|^{12} − |ψ(0)|^{6}|ψ(x)|^{6}dγ^{N} = o(1),

as N → ∞, uniformly in 0 ≤ x < 1/N. This implies that the term (13) makes no contribution when N → ∞. The estimation of (14-18) is similar.

(10) can be overestimated by

(19)  ∥f∥_{∞} ∫ |χ_{K}^{N} − χ_{K}|dγ^{N}.

Now, we have

χ_{K}^{N} − χ_{K} = [h(NH_{1}^{N}) − h(H_{1})] h(−N^{3}H_{3}^{N}) + [h(−NH_{3}^{N}) − h(H_{3})] h(H_{1}).

Therefore, (19) can be overestimated by

(20)  ∴ ≤∥f∥_{∞} ∫ |h(NH_{1}^{N}) − h(H_{1})|h(−N^{3}H_{3}^{N})dγ^{N}
(21)  +∥f∥_{∞} ∫ |h(−N^{3}H_{3}^{N}) − h(H_{3})|h(H_{1})dγ^{N}.

To estimate (20) we note that |ψ(n/N)| ≤ c_{2}(K) on the support of h(−N^{3}H_{3}^{N}) and that

NH_{1}^{N} = −N log ∏(1 − N^{−2}|ψ(n/N)|^{2})

= N^{−1} ∑ |ψ(n/N)|^{2} + \frac{N^{−3}}{2!} ∑ f^{(ii)}|ψ(n/N)|^{4}.

Therefore,

(20) ≤ ∥f∥_{∞}∥h′∥_{∞} ∫_{|ψ(n/N)|≤c_{2}(K)} |NH_{1}^{N} − H_{1}| dγ^{N}

≤ ∥f∥_{∞}∥h′∥_{∞} ∫_{|ψ(n/N)|≤c_{2}(K)} |N^{−1} ∑ |ψ(n/N)|^{2} − ∫_{0}^{1} |ψ|^{2} dx| dγ^{N}

+∥f∥_{∞}∥h′∥_{∞} ∫_{|ψ(n/N)|≤c_{2}(K)} |∑ \frac{f^{(ii)}}{2!} N^{−3}|ψ(n/N)|^{4}| dγ^{N}.

The first term vanishes as in the estimate for the term (20) and the second term vanishes as in the estimate for the term (21).

To estimate the term (21), note that for any ϵ > 0,

γ^{N} (∥ψ∥_{∞} ≥ C) ≥ 1 − ϵ,

for all N provided C is sufficiently large. Then,

(21) ≤ 2ϵ + ∫_{∥ψ∥_{∞}≤C} |h(−N^{3}H_{3}^{N}) − h(H_{3})|h(H_{1})dγ^{N},

in which the integral vanishes for N → ∞ as for (20).

(iv). For any bounded continuous f

∫ fχ_{K}(ψ)e^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄) → ∫ fe^{− \frac{1}{2} W(ψ)}dγ(ψ, ψ̄),

as K → ∞, by the bounded convergence theorem. This implies Ξ_{K} → Ξ as K → ∞. Now the last statement is proved. □

10. Tightness of measures dM_{K}^{N}. We assume that the initial data ψ_{N}(x) has distribution dμ_{N}^{K}. The solutions of initial value problem ψ(x, t) = e^{tX_{3}^{N}} ψ_{N}(x), (x, t) ∈ 𝕊 × [0, T], are realisations of the complex random field with distribution dM_{K}^{N}. To prove tightness of measures we need

Lemma 9. (Kolmogorov—Čentzov). [Ku]. Let the family of complex continuous random fields ϕ(x, t), (x, t) ∈ 𝕊 × [0, T], with distributions dm^{N} satisfy

(1)  E^{N}|ϕ(x, t)|^{γ} ≤ C

and

(2)  E^{N}|ϕ(x_{1}, t_{1}) − ϕ(x_{2}, t_{2})|^{γ} ≤ C (|x_{1} − x_{2}|^{α_{1}} + |t_{1} − t_{2}|^{α_{2}}) ,

where α_{1}^{−1} + α_{2}^{−1} < 1 and γ > 0. Then the family of measures dm^{N} is tight with respect to the weak topology of C (ϕ : 𝕊 × [0, T] → ℂ^{1})

Since realisations of the field are continuously differentiable in the spatial variable x, we consider the random fields ψ′(x, t). The main result of this section is

Lemma 10. For any n = 1, 2, …,:

(3) E_{K}^{N}|ψ′(x, t)|^{2n} ≤ c(n, K)
(4) E_{K}^{N}|ψ′(x_{1}, t_{1}) − ψ′(x_{2}, t_{2})|^{2n} ≤ c(n, K) (|x_{1} − x_{2}|^{n} + |t_{1} − t_{2}|^{n/2})

where E_{K}^{N} is expectation with respected to the measure dM_{K}^{N}.

This, together with Lemma 9, implies tightness of the measures dM′_{K}^{N}. Tightness of the family dM_{K}^{N} can be proved along the same lines. The distribution of the limiting stationary random field we denote by dM_{K}. It follows from the previous section, that the measure dM_{K} has marginal distributions dμ_{K}. In order to establish the Hölder continuity of the spatial derivatives of the random field we need another Lemma 11. (Kolmogorov–Čentzov). [Ku]. Let ϕ(x, t), (x, t) ∈ 𝕊 × [0, T], be a complex random field. Assume that there exist positive constants γ, C, α_{1} and α_{2} with α_{1}^{−1} + α_{2}^{−1} < 1 satisfying

E|ϕ(x_{1}, t_{1}) − ϕ(x_{2}, t_{2})|^{γ} ≤ C (|x_{1} − x_{2}|^{α_{1}} + |t_{1} − t_{2}|^{α_{2}}) .

Then the random field ψ(x, t) has a continuous modification. Moreover c_{0} ≡ (1 − α_{1}^{−1} − α_{2}^{−1})/γ, if β_{1} and β_{2} are positive numbers less then α_{1}c_{0} or α_{2}c_{0} respectively, then there exists a positive random constant C with EC^{γ} < ∞ such that

|ϕ(x_{1}, t_{1}) − ϕ(x_{2}, t_{2})| < C (|x_{1} − x_{2}|^{β_{1}} + |t_{1} − t_{2}|^{β_{2}}) .

By Fatou’s lemma, passing to the limit N → ∞ in the inequality (4), we have

E_{K}|ψ′(x_{1}, t_{1}) − ψ′(x_{2}, t_{2})|^{2n} ≤ C(n) (|x_{1} − x_{2}|^{n} + |t_{1} − t_{2}|^{n/2})

Now Lemma 11 implies that the measure dM′_{K} is supported on paths satisfying

|ψ′(x_{1}, t_{1}) − ψ′(x_{2}, t_{2})| ≤ (|x_{1} − x_{2}|^{1/2−} + |t_{1} − t_{2}|^{1/4−}) .

One can prove, using methods of [V], that the Hölder exponents 1/2− and 1/4− are optimal.

Proof of Lemma 10. To obtain (3), observe that by the invariance of dμ_{K}^{N}

E_{K}^{N}|ψ′(x, t)|^{2n} = E_{K}^{N}|ψ′(x, 0)|^{2n}

≤ \frac{e^{\frac{K}{N}} e^{\frac{c_{4}}{2}}}{inf_{N} Ξ_{K}^{N}} ∫_{M} |ψ′(x, 0)|^{2n}dγ^{N} by Lemma 7

Due to Lemma 8, Ξ_{K}^{N} → Ξ_{K} > 0, as N → ∞. Using the Gaussian character of the measure dγ^{N} the last integral can be overestimated by

≤ c_{1}(K, n) [ ∫_{M} |ψ′(0)|^{2}dγ^{N} ]^{n} = c_{2}(K, n) [ ∑_{|k|≤m} k^{2}σ_{N}^{−2}(k)]^{n}

Now the estimate ak^{4} + 1 ≤ σ_{N}^{2}(k) (see proof of Lemma 6) implies (3). To prove (4), write the differential equation for the flow on M_{N} as in

i\frac{∂ψ(x)}{∂t} = −N^{2}(ψ(x + \frac{1}{N})+ψ(x − \frac{1}{N}) − 2ψ(x))
+|ψ(x)|^{2}(ψ(x − \frac{1}{N}) + ψ(x + \frac{1}{N})),

44 K.L. VANINSKY

where ψ(x) = ψ_{N}(x, t) ∈ M_{N}, x ∈ 𝕊_{N}; equivalently,

e^{tX_{3}^{N}} [ψ(•)](x) = e^{itΔ_{N}} [ψ(•)] (x)
(5)  −i ∫_{0}^{t} e^{i(t−s)Δ_{N}} [|ψ(•, s)|^{2} (ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]_{N} (x)ds,

where

e^{itΔ_{N}} [ψ(•)](x) ≡ ∑_{|k|≤m} e^{2πikx}e^{itΔ(k,N)}ψ̂(k), Δ(k, N) = N^{2}(ω^{k} + ω^{−k} − 2)

and

ψ(x) = ∑_{|k|≤m} e^{2πikx}ψ̂(k).

Since (5) holds for all x ∈ 𝕊_{N} with left and right being polynomials, then (5) holds for all x ∈ 𝕊.

First, we derive the estimate for spatial increments. Much as before,

E_{K}^{N}|ψ′(x + h, t) − ψ′(x, t)|^{2n}
= E_{K}^{N}|ψ′(x + h) − ψ′(x)|^{2n}
≤ c_{2}(K, n) [E_{γ^{N}} |ψ′(h) − ψ′(0)|^{2}]^{n} .

Using the spectral representation of the Gaussian measure γ^{N} in the form

ψ(x) = ∑_{|k|≤m} e^{2πikx}ψ̂(k), E_{γ^{N}} |ψ̂(k)|^{2} = σ_{N}^{−2}(k),

we obtain

E_{γ^{N}} |ψ′(h) − ψ′(0)|^{2} = ∑_{k≠0} 4π^{2}k^{2}|e^{2πikh} − 1|^{2}σ_{N}^{−2}(k).

The estimate |e^{i2πx} − 1| ≤ cx, 0 ≤ |x| ≤ \frac{1}{2}, implies

≤ c_{3} ∑_{0<|k|<h^{−1}/2} \frac{k^{4}h^{2}}{ak^{4} + 1} + c_{3} ∑_{h^{−1}/2≤ |k|} \frac{k^{2}}{ak^{4} + 1}

≤ c_{4}h^{2} ∑_{0<|k|< h^{−1}/2} 1 + c_{4} ∑_{h^{−1}/2≤ |k|} \frac{1}{k^{2}} ≤ c_{5}h.

AN ADDITIONAL GIBBS’ STATE 45

Finally,

(6)  E_{K}^{N}|ψ′(x + h, t) − ψ′(x, t)|^{2n} ≤ c_{6}(K, n)h^{n}.

To obtain the estimate for temporal increments we use (5) and write*

E_{K}^{N}|ψ′(x,t + h) − ψ′(x, t)|^{2n} = E_{K}^{N}|ψ′(x, h) − ψ′(x, 0)|^{2n}
(7)  ≤c_{7} E_{K}^{N} |e^{ihΔ_{N}} ψ′(x) − ψ′(x)|^{2n}
(8)  +c_{7} E_{K}^{N} |∫_{0}^{h} e^{i(h−s)Δ_{N}} [|ψ(•)|^{2} (ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]′_{N} (x)ds|^{2n}

Here

(7) ≤ c_{8} E_{γ^{N}} |e^{ihΔ_{N}} ψ′(x) − ψ′(x)|^{2n} = c_{9} [E_{γ^{N}} |e^{ihΔ_{N}} ψ′(x) − ψ′(x)|^{2}]^{n} . Now, from the spectral respresentation,

E_{γ^{N}} |e^{ihΔ_{N}} ψ′(0) − ψ′(0)|^{2} = ∑_{|k|≤m} |e^{ihN^{2}(ω^{k}+ω^{−k}−2)} − 1|^{2}4π^{2}k^{2}σ_{N}^{−2}(k)
(9)  = ∑_{|k|≤m} (1 − cos hN^{2}(ω^{k} + ω^{−k} − 2))^{2} 4π^{2}k^{2}σ_{N}^{−2}(k)
(10)  + ∑_{|k|≤m} sin^{2} hN^{2}(ω^{k} + ω^{−k} − 2)4π^{2}k^{2}σ_{N}^{−2}(k).

First, we note that for |x| ≤ π

(11)  c_{10}x^{2} ≤ 1 − cos x ≤ c_{11}x^{2}

Therefore, for |k| ≤ m, we have

(12)  c_{12}k^{2} ≤ −N^{2}(ω^{k} + ω^{−k} − 2) ≤ c_{13}k^{2}.

To estimate (8), we assume that |k| ≤ h^{−1/2}α, where α is a suitably chosen constant. Then, by (12),

|hN^{2}(ω^{k} + ω^{−k} − 2)| ≤ π.

*e^{itΔ_{N}} ψ(x) = e^{itΔ_{N}} [ψ(•)] (x)

46 K.L. VANINSKY

Using (11) and (12), we find

(9) ≤ c_{14} ∑_{\begin{matrix}|k|≤m \\ |k|≤h^{−1/2}α \end{matrix}} \frac{(hN^{2}(ω^{k} + ω^{−k} − 2))^{4}}{k^{2} + 1} + c_{15} ∑_{\begin{matrix}|k|≤m \\ |k|>h^{−1/2}α \end{matrix}} \frac{1}{k^{2} + 1}

≤ c_{16}h^{4} ∑_{|k|≤h^{−1/2}α} k^{6} + c_{17} ∑_{|k|>h^{−1/2}α} \frac{1}{k^{2}} ≤ c_{18}h^{1/2}.

The estimate of (10) is even simpler:

(10) ≤ c_{19} ∑_{\begin{matrix}|k|≤m \\ |k|≤h^{−1/2} \end{matrix}} \frac{h^{2}k^{4}}{k^{2} + 1} + c_{20} ∑_{\begin{matrix}|k|≤m \\ |k|>h^{−1/2} \end{matrix}} \frac{1}{k^{2} + 1}

≤ c_{21}h^{2} ∑_{|k|≤h^{−1/2}} k^{2} + c_{22} ∑_{|k|>h^{−1/2}} \frac{1}{k^{2}} ≤ c_{23}h^{1/2}.

Finally, for (7) we have the estimate

(13)  E_{K}^{N}|e^{itΔ_{N}} ψ′(x) − ψ′(x)|^{2n} ≤ c_{24}(K, n)h^{n/2}.

To estimate (8), observe that

E_{K}^{N} |∫_{0}^{h} e^{i(h−s)Δ_{N}} [|ψ(•, s)|^{2}(ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]′_{N} (x)ds|^{2n}

≤ h^{2n−1} ∫_{0}^{h} E_{K}^{N} |e^{i(h−s)Δ_{N}} [|ψ(•, s)|^{2}(ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]′_{N} |^{2n} ds

≤ h^{2n−1} ∫_{0}^{h} E_{K}^{N} |e^{isΔ_{N}} [|ψ(•, s)|^{2}(ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]′_{N} |^{2n} ds.

We will show that, for any s ≥ 0,

(14)  E_{K}^{N} |e^{itΔ_{N}} [|ψ(•)|^{2}(ψ(• + \frac{1}{N} , s) + ψ(• − \frac{1}{N} , s))]′_{N} |^{2n} ≤ c_{25}(K, n).

Then,

(15)  (8) = E_{K}^{N} | … ditto … |^{2n} ≤ c_{25}h^{2n}.

AN ADDITIONAL GIBBS’ STATE 47

The estimates (13) and (15) produce

E_{K}^{N}|ψ′(x, t + h) − ψ′(x, t)|^{2n} ≤ c_{26}(K, n)h^{n/2}.

This and (6) produce (4).

To prove (14) first we note, is ψ(x) = ∑ ψ̂(k)e^{2πikx}, then

ψ_{N}(x) = ∑_{|k|≤m} ψ̂_{N}(k)e^{2πikx},  with  ψ̂_{N}(k) = ∑_{n≡k, (mod N)} ψ̂(n).

To simplify notation we write ψ = ψ_{N}, then

|ψ(x)|^{2} (ψ(x + \frac{1}{N}) + ψ(x − \frac{1}{N}))

= ∑_{|k_{i}|≤m}e^{2πi(k_{1}+k_{3}−k_{2})x}2 cos \frac{2πk_{3}}{N} ψ̂(k_{1})¯ψ̂(k_{2})ψ(k_{3})

and

e^{itΔ_{N}} [|ψ(•)|^{2} (ψ(• + \frac{1}{N}) + ψ(• − \frac{1}{N}))]′_{N} (x)
= ∑_{|k_{i}|≤m}e^{itΔ(N,p)}e^{2πip(k_{1}+k_{3}−k_{2})x}2 cos \frac{2πk_{3}}{N} 2πip(k_{1} + k_{3} − k_{2})ψ̂(k_{1})¯ψ̂(k_{2})ψ̂(k_{3}),

where p(k) ≡ k (mod N) and |p(k)| ≤ m.

Now,

|∂_{x}e^{itΔ_{N}} [|ψ(•)|^{2} (ψ(• + \frac{1}{N}) + ψ(• − \frac{1}{N}))]′_{N} (x)|^{2}
= ∑_{|k_{i}|≤m} e^{it[Δ(N,p)−Δ(N,p′)]}e^{2πix(p(k_{1}+k_{3}−k_{2})−p(k_{4}+k_{6}−k_{5}))}2 cos \frac{2πk_{3}}{N} 2 cos \frac{2πk_{6}}{N}

× 4π^{2}p(k_{1} + k_{3} − k_{2})p′(k_{1} + k_{3} − k_{5})ψ̂(k_{1})¯ψ̂(k_{2})ψ̂(k_{3})¯ψ̂(k_{4})ψ̂(k_{5})¯ψ̂(k_{6}). Estimating

E_{γ^{N}} |… ditto …|^{2n} = |E_{γ^{N}} | … ditto … |^{2n}|

= | ∑_{j=1}^{n} ∏ e^{it[Δ(N,p^{j})−Δ(N,p′^{j})]}e^{2πix(p(k_{1}^{j}−k_{3}^{j}−k_{2})−p(k_{4}^{j}+k_{6}^{j}−k_{5}^{j}))}2 cos \frac{2πk_{3}^{j}}{N} 2 cos \frac{2πk_{6}^{j}}{N}

× E_{γ^{N}} ∏ 4π^{2}p (k_{1}^{j} + k_{3}^{j} − k_{2}^{j})p′(k_{4}^{j} + k_{6}^{j} − k_{5}^{j})ψ̂(k_{1}^{j})¯ψ̂(k_{2}^{j})ψ̂(k_{3}^{j})

× ¯ ψ̂(k_{4}^{j})ψ(k_{5}^{j})¯ψ̂(k_{6}^{j})|

≤ ∑_{j=1}^{n} E_{γ^{N}} ∏_{|k_{i}^{j}|≤m} 4π^{2}|p(k_{1}^{j} + k_{3}^{j} − k_{2}^{j})||p(k_{4}^{j} + k_{6}^{j} − k_{5}^{j})

× ψ̂(k_{1}^{j})¯ψ̂(k_{2}^{j})ψ̂(k_{3}^{j})¯ψ̂(k_{4}^{j})ψ̂(k_{5}^{j})¯ψ̂(k_{6}^{j}).

Using the inequalities

|p(k_{1} + k_{3} − k_{2})| ≤ |k_{1} + k_{2} − k_{3}| ≤ (1 + |k_{1}|)(1 + |k_{2}|)(1 + |k_{3}|),

and

(1 + |k|)^{2} ≤ 2(1 + |k|^{2}),

we overestimate the last sum by

≤ c_{27} ∑_{|p_{i}|≤m} E_{γ^{N}} (1 + p_{1}^{2})|ψ̂(k_{1})|^{2} … … (1 + p_{1}^{2})|ψ̂(p_{3n})|^{2}.

The worst term

∑ E_{γ^{N}} p_{1}^{2}|ψ̂(p_{1})|^{2} . . . . . . p_{3n}^{2}|ψ̂(3n)|^{2}.

can be overestimated by

∴≤ E_{γ^{N}} |ψ′(x)|^{2×3n} ≤ C(n, N)

due to ak^{4} + 1 ≤ σ_{N}^{2}(k). □

11. Identification of measure dM_{K}. This section shows that the measure dM_{K} is supported on the solutions of the NLS flow.

Multiplying both parts of the original equation

iψ^{•} = −ψ′′ + 2|ψ|^{2}ψ

by the test function f(x, t), (x, t) ∈ 𝕊 × (0, T] and integrating produces

∫_{0}^{1} ∫_{−∞}^{+∞} dx dt [iψf^{•} − ψf′′ + |ψ|^{2}2ψf] = 0.

Lemma 12. In the statistical ensemble dM_{K}

E_{M_{K}} |∫_{0}^{1} ∫_{−∞}^{+∞} dxdt [iψf^{•} − ψf′′ + |ψ|^{2}2ψf]|^{2} = 0.

Proof. The equation for the AL flow on M_{N} is

iψ^{•}(x, t) = −Δ_{N}ψ(x, t) + |ψ(x, t)|^{2}(ψ(x + \frac{1}{N} , t) + ψ(x − \frac{1}{N} , t)),

where Δ_{N}ψ(x) = N^{2} (ψ(x + \frac{1}{N}) + ψ(x − \frac{1}{N}) − 2ψ(x)) and x ∈ 𝕊_{N}. We multiply by the test function f(x, t) and integrate over the variables t and x.

∫_{0}^{1} ∫_{−∞}^{+∞} dx dt δ_{N}(x)[iψ(x, t)f^{•} − ψ(x, t)Δ_{N}f
+|ψ(x, t)|^{2} (ψ(x + \frac{1}{N} , t) + ψ(x − \frac{1}{N} , t)) f] = 0,

in which δ_{N}(x) = \frac{1}{N} ∑_{x_{0}∈𝕊_{N}} δ(x − x_{0}). Now

E_{M_{K}} |∫ ∫ dx dt [iψf^{•} − ψf′′ + |ψ|^{2}2ψf]|^{2}

= lim_{N→∞}E_{M_{K}^{N}} |... ditto ...|^{2}
= lim_{N→∞}E_{M_{K}^{N}} | ∫ ∫ dx dt iψf^{•}(1 − δ_{N}(x)) − ∫ ∫ dx dt ψ(∂_{xx}^{2} − δ_{N}Δ_{N})f

+ ∫ ∫ dx dt [|ψ|^{2}2ψ − |ψ|^{2} (ψ(x + \frac{1}{N}) + ψ(x − \frac{1}{N})) δ_{N}(x)] f|^{2}

≤ lim_{N→∞} cE_{M_{K}^{N}} |∫ ∫ dx dtψf^{•}(1 − δ_{N})|^{2}
+ lim_{N→∞} cE_{M_{K}^{N}} |∫ ∫ dxdtψ(δ_{xx}^{2} − δ_{N}Δ_{N})f|^{2}

+ lim_{N→∞} cE_{M_{K}^{N}} |∫ ∫ dx dt [|ψ|^{2}2ψ − |ψ|^{2} (ψ(x + \frac{1}{N}) + ψ(x − \frac{1}{N})) δ_{N}(x)] f|^{2} .

The first term overestimated by

∴≤|f^{•}|_{∞}E_{M_{K}^{N}} (∫_{0}^{1} ∫_{0}^{T} dx dt|ψ(x, t) − ψ(k/N, t)|dx)^{2}

≤|f^{•}|_{∞}T ∫_{0}^{T} E_{M_{K}^{N}} (∫_{0}^{1} |ψ(x, t) − ψ(k/N, t)|dx)^{2}

≤|f^{•}|_{∞}T^{2} ∫_{0}^{1} dxE_{M_{K}^{N}} |ψ(x, t) − ψ(k/N, t)|^{2}

=|f^{•}|_{∞}T^{2}N ∫_{0}^{1/N} E_{M_{K}^{N}} |ψ(x) − ψ(0)|^{2}dx
≤|f^{•}|_{∞}T^{2} sup_{0≤x≤1/N}E_{M_{K}^{N}} |ψ(x) − ψ(0)|^{2} = o(1),

as N → ∞ due to the stochastic continuity of the random field ψ. Estimates for the remaining terms can be obtained along the same lines. □

Due to the results of [MCV1, B2] the initial data ψ(x, t) determines the flow ψ(•, t) for all t. In probabilistic language this means that ψ(•, t) is measurable with respect to the field generated by ψ(•, 0).

References

[AL] M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations and Fourier analysis, Journ. Math. Phys. 17 (1976), 1011-1018.

[B1] J. Bourgain, Periodic Nonlinear Schrödinger Equation and Invariant Measures, Comm. Math. Phys. (1994), no. 166, 1-26.

[B2] J. Bourgain, Global solutions of Nonlinear Schrödinger Equations, Colloquium Publications, vol. 46, Amer. Math. Soc., Providencece, Rhode Island, 1999.

[KP] I. Krichever and D.H. Phong, Symplectic Forms in the Theory of Solitons, hepth/9708170.

[Ku] H. Kunita, Stohastic Flows and Stohastic Differential Equation, Cambridge University Press, Cambridge, 1990.

[MC] H.P. McKean, Statistical Mechanics of Nonlinear Wave Equations (4) Cubic Schrödinger, Comm. Math. Phys. (1995), no. 168, 479–491; Comm. Math. Phys. (1995), no. 173, 675.

[MCV1] H.P. McKean and K.L. Vaninsky, Action-angle variables for nonlinear Schrödinger equation, Comm. Pure Appl. Math. 50 (1997), 489-562.

[MCV2] H.P. McKean and K.L. Vaninsky, Cubic Schrödinger: The Petit canonical Ensemble in Action–Angle variables, Comm. Pure and Appl Math. 50 (1997), 593-622.

[MEKL] P.D. Miller, N.M. Ercolani, I.M. Krichever, C.D. Levermore, Finite Genus Solutions to the Ablowitz-Ladik Equations, Comm. Pure. Appl. Math. 48 (1995), 1369–1440.

[V] K.L. Vaninsky, On Space-time Properties of Solutions for Nonlinear Evolutionary Equations with Random Initial Date, Revista de Matematica: Theoria y Aplicaciones 3 (1996), no. 1, 11-20.