Vessels
Most of the result presented here can be found here.
Definition: Collection of operators and spaces
V = (A, B(x), X(x); σ1, σ2, γ, γ*(x); H, Cp; I),
where H is a Hilbert space; A, X(x)= X*(x): H →H , B(x): Cp →H are linear operators, σ1 = σ1*, σ2 = σ2*, γ = - γ*, γ* (x) = - (γ*(x))* are pxp matrices, is called a vessel if the following conditions hold:
· (AReg) the operator A has a dense domain D (A) with a symmetric spectrum with respect to the imaginary axis. A is the generator of a C0 semi-group on H, R(λ) = (λ I - A)-1is the Resolvent.
· B(x) satisfies regularity assumptions (for all x in I and all λ out of the spectrum of A)
(Bσ2) the image of R (λ) B(x) σ2 is in D (A),
(Bγ) the image of R (λ) B(x) γ is in H.
B(x) also satisfies
(ResDB) 0 = δ/δx [R(λ) B(x)] σ1 + A R(λ) B(x) σ2 + R(λ) B(x) γ.
· X(x) is a bounded, self-adjoint, invertible on some interval I and satisfies
(DX) d/dx X(x) = B(x) σ2 B*(x).
· The Lyapunov equation holds for all x in I and all λ out of the spectrum of A:
(ResLyapunov) X(x) R*(- λ*) + R(λ) X(x) - R(λ) B(x) σ1 B*(x) R*(- λ*) = 0.
· γ*(x) satisfies the linkage condition on I
(Linkage) γ*(x) = γ+ σ2 B*(x) X-1(x)B(x) σ1- σ1 B*(x) X-1(x)B(x) σ2.
Using these assumptions it is possible to show that the basic results from the theory of regular vessels hold: the permanency of the Lyapunov equation, The Bäcklund transformation theorem, and the formula for γ*(x) in terms of the tau function in SL case. But also a very interesting notion of a pre-vessel also arises from this construction:
Pre-vessels and generalized inverse scattering
Definition: Collection of operators and space
preV = (A, B(x), X(x); σ1, σ2, γ; H, Cp; I),
where A, X(x)= X*(x): H →H , B(x): Cp →H are linear operators, σ1= σ1*, σ2= σ2*, γ=- γ* are pxp matrices, is called a pre-vessel if the conditions (AReg), (Bσ2), (Bγ), (ResDB),(DX), (ResLyapunov) hold.
Notice that these are all the condition of the vessel, except for the invertability of X and the definition of γ*(x). If the operator X(x) is invertible for a given x, then we can define the generalized potential γ*(x). Notice that such a potential exists for all points of the real line, where this happens (or equivalently, where the tau function is non-zero). Thus we obtain the following
Definition: The generalized potential of a pre-vessel preV is the function γ*(x), defined by (Linkage) for all points where τ(x) is non-zero.
Problem [Generalized scattering theory] Characterize the class GP of generalized potentials, arising from pre-vessels.
Problem [Generalized inverse scattering] Given a generalized potential γ*(x) in GP, find a pre-vessel realizing it.
For example, using Gelfand-Levitan theory, in a special case below, we obtain
Theorem: Let q(x) (0 < x < ∞) be a potential for which the function f(x,y) possesses the form (Deff). Then there exists a pre-vessel, whose generalized potential coincides with q(x) on the positive real line.
Construction of a SL vessel on the half line using a Gelfand-Levitan theory
Let us show how to construct a vessel for SL vessel parameters. Following a Gelfand-Levitan theory, suppose that q(x) is a continuously differentiable potential, defined on the positive half line (R+). Then the Sturm-Liouville differential equation with the spectral parameters λ and initial conditions
(SL) - d2/dx2 y(x) + q(x)y(x) = λ y(x), y(0)=1, y’(0)=0
Possesses a spectral function ρ (λ), which is a monotone function on R. Their main result is as follows
Theorem Let ρ (λ) be a monotone function. Let
ω (λ) = ┌ ρ (λ) – 2/π λ, λ > 0
└ ρ (λ), λ < 0
and suppose that ω (λ) satisfies (for all x in R+)
1* ∫0-∞ exp(√|λ| x) d λ < 0,
2* a(x) = ∫∞1 cos(√|λ| x)/ √|λ| d λ has fourth continuous derivative.
Then there exists a continuous potential q(x), corresponding to the spectral function ρ (λ). Conversely, if q(x) is continuously differentiable, there exists a spectral function ρ (λ) possessing properties 1*, 2*.
Proof: see GL paper (Russian) □
We use SL vessel parameters σ1, σ2, γ (see regular vessels). Suppose that the function ω (λ) is monotone increasing and satisfies
(Deff) f(x,y) = ∫R cos(x√λ)cos(x√λ) d ω (λ) < ∞
for all x, y. Then define H = L2 (d ω (λ)), A = i λ and the following operators
B(x) = B(x, μ) = [cos(x √μ) & i √μ sin(x √μ)]
The operator B(x) acts on C2 as a multiplication by the row matrix B(x, μ). Define next operator X(x): H →H using an arbitrary g in H. Here we present how to calculate the resulting function X(x) g at the point λ:
(X(x) g λ) = [I + ∫0x B(y) σ2 B*(y) dy ]g = g(λ) + ∫0x cos (x√λ) ∫R cos (y√λ) g(μ) d ω (μ)
Theorem the collection V = (A, B(x), X(x); σ1, σ2, γ, γ*(x); H, C2; I) for the spaces and operators defined above is a vessel.
Then the ingredients of the Gelfand Levitan equation are expressed as follows
Construction of a SL vessel on the line
Using results of Fadeyev, Agronovich-Marchenko we can construct a vessel corresponding to a potential. This was done here.