Regular Symmetric Vessels

Definition: Collection of operators and space

V = (A, B(x), X(x); σ₁, σ₂, γ, γ_*(x); H, C^p; I),

where A, X(x)= X*(x): H →H , B(x): C^p →H are differentiable bounded operators, σ₁₌ σ₁*, σ₂₌ σ₂*, γ=- γ*, γ_* (x)= - (γ_*(x))* are pxp matrices,is called a regular symmetric vessel if the following conditions hold:

(DB) d/dx [B (x) σ₁] + A B(x) σ₂ + B(x) γ = 0,

(DX) d/dx X(x) = B(x) σ₂ B*(x), X(x) is invertible on I,

(Linkage) Linkage condition: γ_*(x) = γ+ σ₂ B*(x) X^-1(x)B(x) σ₁- σ₁ B*(x) X^-1(x)B(x) σ₂

(Lyapunov) Lyapunov equation: A X(x) + X(x)A*+ B(x) σ₁ B*(x)=0

Remark: “regular” means all the involved operators are bounded, which simplifies the formulas and proves. The theory of vessels with an unbounded operator A can be found here. As a background, regular vessels should be understood first.

Standard construction of a regular symmetric vessel

Let A, X₀=X₀^*: H →H, B₀: C^p →H be operators satisfying

· Lyapunov equation: A X₀ + X₀ A*+ B₀σ₁ B₀*=0,

· X₀ is invertible

Then

_1. Define B(x) as the unique solution of (DB) with the initial value B_0,

2. Define X(x) as the unique solution of (DX) with the initial value X₀. It will be invertible on an interval I, since it is invertible at 0 and is continuous,

3. Define γ_*(x) on I using (Linkage).

Theorem [permanency of the Lyapunov equation]: if it holds at any point it holds for all points.

Proof: using (DB), (DX) the derivative of left hand side of (Lyapunov) is zero □

Theorem: γ_*(x) of a regular vessel is analytic on I.

Proof: immediate from (Linkage) since B(x), X^-1(x) are analytic on I □

Some choices of parameters:

I – identity matrix, 0 – zero matrix, i=√-1.

Tau function

Definition: Tau function of the vessel V is defined as follows

τ(x) = det((X₀)^-1X(x))

It plays a crucial role in the vessel existence: vessel exists on I where X(x) is invertible.

From the definition it follows that τ’(x)/ τ(x)=Tr(X’(x)X^-1(x)) and it is possible to show that γ_*(x) has a special form, related to τ(x)

Transfer function

Definition: Transfer function of the vessel V is defined as follows

(Realize) S(λ,x)=I - B*(x) X^-1(x)( λ I - A)^-1 B(x) σ₁

defined for λ out the spectrum of A and for x in I.

Theorem: Lyapunov equation (Lyapunov) implies the symmetry of S(λ,x):

(Sym) S* (-λ*,x) σ₁S(λ,x)= σ₁

Proof: Plugging the realization formula (Realize) into (Sym) and using (Lyapunov) □

Theorem [Vessel = Bäcklund transformation]: Let S(λ,x) be the transfer function of a vessel V. If u(λ,x) satisfies the input compatibility equation

(InputCC) [- σ₁ δ/δx + σ₂ λ + γ] u(λ,x) = 0

Then y(λ,x)= S(λ,x) u(λ,x) satisfies the output compatibility condition

(OutCC) [- σ₁ δ/δx + σ₂ λ + γ_*(x)] u(λ,x) = 0

Proof: Plug the realization formula (Realize) for S(λ,x) into y(λ,x). Use (DB), (DX) and (InputCC) to derive (OutCC) □

From here it follows that S(λ,x) satisfies the following differential equation

(DS) δ/δx S(λ,x) = σ₁^-1 (λσ₂ + γ_*(x)) S(λ,x) - S(λ,x) σ₁^-1 (λσ₂ + γ)

Theorem [Realization]: If S(λ,x) is symmetric, analytic in λ at infinity, analytic in x on I, multiplication by S(λ,x) maps solutions of (InputCC) to solutions of (OutCC) then there exists a regular vessel V whose transfer function coincides with the given S(λ,x).

Proof: First using theorems on symmetric functions, analytic at infinity, we can realize S(λ,x₀). Then using this realization we apply the standard construction to obtain a vessel with transfer function Y(λ,x). Then S(λ,x) Y^-1(λ,x) is entire analytic at infinite, hence by Liouville theorem S(λ,x) Y^-1(λ,x) is constant, namely identity (the value at infinity) □

Moments

From the definition of the transfer function, taking the Taylor expansion around infinity, we can obtain that

(SMomnts) S(λ, x) = I - ∑ H_n(x) σ₁ / λⁿ⁺¹

using

Definition: n-th moment of the vessel V is defined as follows

(DefH) H_n(x) = B* X^-1(x) Aⁿ B(x), n = 0, 1, 2, 3,…

Theorem: The moments satisfy the following recurrence relations:

(MDRecrs) σ₁^-1 σ₂ H_n+1(x) - H_n+1(x) σ₁^-1 σ₂ = d/dx H_n(x) - σ₁^-1 γ_*(x) H_n(x) + H_n(x) σ₁^-1 γ

(MARecrs) H_n+1(x) σ₁^-1 + (-1)ⁿ σ₁^-1 H_n+1(x)* = ∑_j=0ⁿ (-1)^j+1 H_n-j(x) σ₁^-1 H_j*

Proof: To obtain the first equation (MDRecrs), plug the expansion (SMoments) into the differential equation (DS) and compare the coefficients of the powers of λ. Similarly, use (Sym)for the second equation (MARecrs). □

Associated Completely Integrable System

There is an integrable system corresponding to the vessel. Before we present the integrable system, we consider a simpler one first, which is directly connected to inputs/outputs defined earlier. Let (u(λ,x)/x(λ,x)/y(λ,x)) be the (input/state/output) triple of the following overdetermined system of equations:

┌ x(λ,x) = (λ I - A)^-1 B(x) σ₁ u(λ,x)

(System) ┤ d/dx x(λ,x) = B(x) σ₂ u(λ,x)

└ y(λ,x) = u(λ,x) – B*(x) X^-1(x) x(λ,x)

Overdetermined means that only u(λ,x) satisfying (InputCC) are considered. Then it follows that

y(λ,x) = S(λ,x) u(λ,x)

Equation (DB) makes the two equations for x(λ,x) be compatible (for the “legal” u(λ,x)). Then equation (DX) makes y(λ,x) to satisfy (OutCC).

The system (System) is derived from a completely integrable 2D system (CI System) by considering “waves“. More precisely, consider the following system

┌ δ/δt x(t,x) = A x(t,x) + B(x) σ₁ u(t,x)

(CI System) ┤ δ/δx x(t,x) = B(x) σ₂ u(t,x)

└ y(t,x) = u(t,x) – B*(x) X^-1(x) x(t,x)

which is overdetermined, namely, only u(t,x) satisfying

(InputCC’) [- σ₁ δ/δx + σ₂ δ/δt + γ] u(λ,x) = 0

is allowed. Then the output satisfies

(OutCC’) [- σ₁ δ/δx + σ₂ δ/δt + γ_*(x)] y(λ,x) = 0

when operators satisfy (DB), (DX) and (Lyapunov). Moreover this system is integrable (for legal inputs!), because the mixed second derivatives are equal:

δ/δt δ/δx x(t,x) = δ/δx δ/δt x(t,x),

which can be checked by the use of (CI System) equations and (DB), (DX), (Lyapunov). Performing here separation of variables as follows

u(t,x)=e^λt u(λ,x), x(t,x)=e^λt x(λ,x), y(t,x)=e^λt y(λ,x)

we will arrive to the first system (System). So, vessels indeed come from completely integrable systems by separation of variables.

Standard model for the continuous spectrum on a bounded/symmetric curve

Suppose that we are given a smooth curve in the complex domain, parameterized by Γ = { μ(t) | a≤ t≤ b } (a<b are finite numbers). Let us also suppose that the curve is symmetric with respect to the imaginary axis, i.e. Γ = - Γ *. The reason why we need to require it is the symmetry condition (Symmetry). The inner space is defined

H = L² (Γ) = {f (μ) | ∫_Γ |f(μ(t))|² dt < ∞}.

with the corresponding inner product

< f, g > _H = ∫_Γ g* (μ(t)) f (μ(t)) dt.

Define the operator A as the multiplication operator on a function μ(t): A f(μ(t)) = μ(t) f(μ(t)). Then B(x) is a solution of (DB) with an initial condition B_0. If we denote the fundamental solution of (InputCC) as Φ(x,λ), then

B(x) = B(μ,x)=[b₁ b₂] Φ(x,λ), for some b₁, b₂ from H

Notice that from the definition it follows that the adjoint of B(x) is

B*(x) f = ∫_Γ B*(μ,x) f(μ(t)) dt.

Define the operator X(x) as follows

(X(x) f)(μ) = ∫_Γ [B (μ,x) σ₁ B*(δ(t),x)]/[ μ + δ (t)] dt.

Notice that in order to obtain a well-defined operator, we have to verify that the integral converges. For this to happen, we can demand that for the values of δ(t), where μ + δ (t)= 0, it holds that the Hölder condition holds for B(μ,x) σ₁ B*(δ,x).

Theorem The collection

V = (A, B(x), X(x); σ₁, σ₂, γ, γ_*(x); H, C^p; I),

For the operators defined above is a regular vessel.

Proof: Equation (DB) is satisfied by the construction of B(μ,x). Lyapunov equation (Lyapunov) follows almost immediately, equation (DX) also is easy to verify. Finally equation(Linkage) serves to define γ_*(x) □

Finally notice that from (Realize) the transfer function of this vessel is

S(λ,x)=I - ∫_Γ B*(δ(t),x) X^-1(x)( λ I - δ(t))^-1 B(δ(t),x) σ₁ dt

and has singularities (jumps) at most on the curve Γ. If we demand additionally that the two functions b₁, b₂ on curve Γ do not vanish together, the transfer function will have jumps exactly on this curve. The vessel exists on I, where X(x) is invertible.

Standard model for the discrete spectrum (on the imaginary axis)

Let H be the space l² = {s_n}. On this space define A = diag(i λ_n²)_, for an arbitrary fixed and bounded sequence of real numbers {λ_n} or more precisely A {s_n} = { i λ_n² s_n}. In order to define the operator B(x) we fix an l² sequence of non-zero entries {b_n} and define:

B(x) = diag(b_n) [cos(λ_n x) -i λ_n sin(λ_n x)]

It is an infinite sequence of rows of two entries. Notice that B*(0) σ₁ B(0) = 0 and it is necessary to choose X(0) so that (Lyapunov) holds or that A X(0) + X(0) A* = 0. Since A is diagonal, we choose X(0) = diag(x_n), where the sequence {x_n} is bounded from above AND from below (because X(0) must be invertible operator). Then integrating B(y) σ₂ B*(y)from 0 to x, we find that from (DX) the (n,m) entry of the operator X(x) is as follows:

X(x) = [cos(λ_n x) λ_m sin(λ_m x)- λ_n sin(λ_n x) cos (λ_m x)]/(λ_n² – λ_m²) if n≠m

(2 λ_nx + sin(2λ_n x))/(4 λ_n) if n=m

Theorem The collection

V = (A, B(x), X(x); σ₁, σ₂, γ, γ_*(x); H, C^p; I),

For the operators defined above is a regular vessel.

Proof: Equations (DB), (Lyapunov), (DX) follows immediately. Equation (Linkage) serves to define γ_*(x) □

The transfer function of this vessel has poles at λ = i λ_n² and at the closure of this set. This follows immediately from the realization formula (Realize).