Wave model of symmetric semibounded operators M. I. Belishev 1 , S. - - PowerPoint PPT Presentation

Wave model of symmetric semibounded operators

• M. I. Belishev1, S. A. Simonov1,2

1: St. Petersburg Department of V. A. Steklov Mathematical Institute, 2: St. Petersburg State University

Functional model

H – a (separable) Hilbert space; A : H → H ;

• H – model space;

U : H → H – a ‘canonical’ unitary operator,

• A = UAU ∗ :
• H →

H (traditionally, H is a function space, A is multiplication by function).

• B. Sz.-Nagy–C. Foias: functional model of contraction
• B. Pavlov: functional model of dissipative operator
• A. Strauss: functional model of symmetric operator

Determined by the characteristic function, the Weyl function, etc.

• M. Belishev (2013): the wave model of symmetric semibounded operator (heuristic level!)

based on the notion of the wave spectrum. Motivation comes from inverse problems of mathematical physics: the WM is determined by the inverse data. PROGRAM: to justify the wave model and turn it into a rigorous functional model of a class of symmetric operators

Plan of the talk

• 1. Green system
• 2. Dynamical system with boundary control. Boundary lattice
• 3. Dual system. Isotony
• 4. Wave spectrum and wave model
• 5. Examples

• 1. Green system

{H , B; L0; Γ1, Γ2} H , B – separable Hilbert spaces, H – inner space, B – boundary space; L0 : H → H – basic operator, Dom L0 = H , L0 = L0, L0 ⊂ L∗

0;

n+[L0] = n−[L0], 1 n±[L0] ∞; Γ1,2 : H → B – boundary operators, Dom Γ1,2 ⊃ Dom L∗

0, Ran Γ1,2 = B.

GREEN FORMULA: (L∗

0u, v)H − (u, L∗ 0v)H = (Γ1u, Γ2v)B − (Γ2u, Γ1v)B

If L0 γ1 with γ > 0, then there is a canonical Green system associated with the I. M. Vishik decomposition Dom L∗

0 ∋ y = y0 + h + L−1g,

where y0 ∈ Dom L0, g, h ∈ Ker L∗

0, L is the Friedrichs extension: L0 ⊂ L ⊂ L∗ 0, L γ1.

Define B := Ker L∗

0, Γ1 : y → −h, Γ2 : y → g. Then {H , B; L0; Γ1, Γ2} is a Green system.

L0 determines a canonical (Vishik) Green system! Example: H = L2(R+), B = C; L0y = −y′′ + q(x)y, Dom L0 = {y ∈ L2(R+) ∩ H2

loc([0, ∞)) | y(0) = y′(0) = 0, −y′′ + qy ∈ L2(R+)},

the potential q is such that L0 γ1; Γ1y = −y(0)φ, Γ2y = y′(0)−y(0)φ′(0)

η′(0)

φ, where φ obeys −φ′′ + qφ = 0, φ(0) = 1, φ ∈ L2(R+) and η = L−1φ.

• 2. Dynamical system with boundary control

Recall that L0 γ1. The DSBC is utt + L∗

0u = 0

in H , t > 0 ; u|t=0 = ut|t=0 in H ; Γ1u = f(t), t 0; f is a boundary control of the ‘smooth’ class M := {f ∈ C∞([0, ∞); B) | f ≡ 0 near t = 0}, u = uf(t) is a solution (wave). For f ∈ M the DSBC has a unique classical solution. Reachable sets: U T := {uf(T) | f ∈ M }, T > 0; U T↑ as T ↑ . Example: the Sturm–Liouville DSBC utt − uxx + qu = 0, x > 0, t > 0; u|t=0 = ut|t=0 = 0, x 0; u|x=0 = f(t), t 0; u = uf(x, t) is the wave. Here U T = L2(0, T) (the speed of waves equals 1!). DSBC is determined by L0!

• 3. Dual system

Recall that L = L∗ is the Friedrichs extension of L0. The system dual to DSBC is vtt + Lv = h in H , t > 0; v|t=0 = vt|t=0 = 0 in H . For h ∈ C∞([0, ∞); H ), h ≡ 0 near t = 0 there a unique classical solution v = vh(t). Reachable sets: for A ∈ L[H ] define V T

A := {vh(T) | h ∈ C∞([0, ∞); A ), h ≡ 0 near t = 0}.

Note that V T

A ↑ as T ↑. Define

IT : L[H ] → L[H ], ITA := V T

A

Monotonicity: if A ⊂ B, T1 T2 then IT1A ⊂ IT2B. Isotony: I := {IT | T 0}, I0 := id. Isotony is determined by L0!

Example: the system dual to Sturm–Liouville DSBC vtt − vxx + qv = h, x > 0, t > 0; v|t=0 = vt|t=0 = 0, x 0; v = vh(x, t) is the solution.

• Lemma. The S.–L. isotony acts as

ITL2(a, b) = L2(max {0, a − T}, b + T), T > 0 . (⋆) Hilbert lattice of subspaces L[H ] (a reminder): Operations: F, G ∈ L[H ] implies F ∧ G := F ∩ G , F ∨ G := span {F, G }, F ⊥ := H ⊖ F ∈ L[H ]. Topology: Fj → F ⇔ s− lim

j→∞ Pj = P, where Pj, P are the orthogonal projections

• n Fj, F.

Boundary lattice of DSBC: the minimal (sub)lattice LU ⊂ L[H ] that obeys U T ∈ LU and ITLU ⊂ LU , T > 0. Boundary lattice is determined by L0! Example: the boundary lattice of Sturm–Liouville DSBC is LU =

• G ∈ L[H ]
• G =

n

• j=1

L2(aj, bj), 0 a1 < b1 < a2 < b2 < · · · < an < bn ∞

• (follows from (⋆)).

• 4. Wave spectrum and wave model

Let F := {f : [0, ∞) → L[H ]} be the lattice of L[H ]-valued functions of time with the point-wise operations (f ∧ g)(t) := f(t) ∧ g(t), (f ∨ g)(t) := f(t) ∨ g(t), (f ⊥)(t) := (f(t))⊥, t 0 and the pointwise convergence fj → f ⇔ fj(t)

L[H ]

→ f(t), t 0. Define FU := {f ∈ F | f(t) = It[f(0)], t 0, f(0) ∈ LU } ⊂ F. Atoms (a reminder): Let P be a partially ordered (by ) set with the lowest element 0. An element a ∈ P is an atom, if 0 < b a implies b = a. We denote by At P the set of atoms of P. Wave spectrum of L0: FU is a partially ordered set with the order ⊆. The wave spectrum of L0 is ΩL0 := At FU (the closure is important!). Distance on ΩL0: d(α, β) = 2 inf{t > 0 | α(t) ∧ β(t) = 0} (interaction time). Wave spectrum is a unitary invariant of L0 !

Wave model: Transform U: elements y ∈ H → functions y = Uy on the wave spectrum by

• y(α) := lim

t→+0

• Pα(t)y, e
• H
• Pα(t)e, e
• H

, α ∈ ΩL0, where α = α(t) is an atom (an H -valued function of time), Pα(t) projects in H onto the subspace α(t) ⊂ H ; e ∈ H is a ‘relevant’ gauge element. Important! The construction of the wave model is fully determined by L0 given in any representation (i.e., by any unitary copy of L0). Available for inverse problems: Inverse Data ⇒                   

• charact. function of L0;

spectral data; Weyl function of the Green system; scattering data; boundary spectral and dynamical data ⇒       Determines L0 up to unitary equivalence (L0 is completely non-selfadjoint)       ⇒ Wave model

• 5. Examples
• 1. H = L2(R+) (R+ = {x 0}), B = C; the Sturm–Liouville operator

L0 = − d2 dx2 + q(x)

• n

Dom L0 =

• y ∈ L2(R+) ∩ H2

loc(R+) | y(0) = y′(0) = 0, −y′′ + qy ∈ L2(R+)

• ,

q is such that L0 γ1 (γ > 0); Dom L = {y ∈ L2(R+) ∩ H2

loc(R+) | y(0) = 0, −y′′ + qy ∈ L2(R+)},

Dom L∗

0 = {y ∈ L2(R+) ∩ H2 loc(R+) | − y′′ + qy ∈ L2(R+)} : L0 ⊂ L ⊂ L∗ 0 .

Wave Model:

• H = L2(R+) (R+ = {τ 0}),

B = C;

• L0 = −κ−1 d2

dτ 2κ + q(τ) (κ is a smooth positive function), with the same q ! Inverse Problem: ρ(λ) – the spectral function of L ⇒ Wave model ⇒ q.

2. Ω a smooth compact n-dimensional Riemannian manifold with the boundary ∂Ω; H = L2(Ω), B = L2(∂Ω); the Beltrami–Laplace operator L0 = −∆

• n Dom L0 = {y ∈ H2(Ω) | y|∂Ω = ∂νy|∂Ω = 0}

(ν is the outward normal). Wave spectrum: ΩL0

isometric

∼ = Ω. Wave model:

• L0 = κ−1L0κ in L2(ΩL0) (κ is a smooth positive function).

Inverse Problem: given the boundary inverse data to recover Ω.

Comparing the models

Sz.-Nagy–Foias Pavlov Strauss   

• perator of multiplication by independent variable

in a model Hilbert space of functions Wave Model: an operator of the same kind as the original operator.

Recent papers

• 1. M. I. Belishev and S. A. Simonov. Wave model of the Sturm–Liouville operator on the

half-line, St. Petersburg Math. J., 29(2), 227–248, 2018.

• 2. S. A. Simonov. The wave model of the Sturm–Liouville operator on an interval, J.
• Math. Sci. (N. Y.), 243(5), 783–807, 2019.
• 3. M. I. Belishev and S. A. Simonov. A wave model of metric spaces, Funct. Anal. Appl.,

53(2), 79–85, 2019.

• 4. M. I. Belishev and S. A. Simonov. A wave model of a metric space with measure, to

appear in Sbornik: Mathematics.

• 5. M. I. Belishev. A unitary invariant of a semi-bounded operator in reconstruction of

manifolds, J. Operator Theory, 69(2), 299–326, 2013.