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

Wave model of symmetric semibounded operators M. I. Belishev 1 , S. A. Simonov 1 , 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 ; L 0 ; Γ 1 , Γ 2 } H , B – separable Hilbert spaces, H – inner space, B – boundary space; L 0 : H → H – basic operator, Dom L 0 = H , L 0 = L 0 , L 0 ⊂ L ∗ 0 ; n + [ L 0 ] = n − [ L 0 ], 1 � n ± [ L 0 ] � ∞ ; Γ 1 , 2 : H → B – boundary operators, Dom Γ 1 , 2 ⊃ Dom L ∗ 0 , Ran Γ 1 , 2 = B . ( L ∗ 0 u, v ) H − ( u, L ∗ GREEN FORMULA: 0 v ) H = (Γ 1 u, Γ 2 v ) B − (Γ 2 u, Γ 1 v ) B If L 0 � γ 1 with γ > 0, then there is a canonical Green system associated with the I. M. Vishik decomposition Dom L ∗ 0 ∋ y = y 0 + h + L − 1 g, where y 0 ∈ Dom L 0 , g, h ∈ Ker L ∗ 0 , L is the Friedrichs extension: L 0 ⊂ L ⊂ L ∗ 0 , L � γ 1 . Define B := Ker L ∗ 0 , Γ 1 : y �→ − h , Γ 2 : y �→ g . Then { H , B ; L 0 ; Γ 1 , Γ 2 } is a Green system. L 0 determines a canonical (Vishik) Green system! Example: H = L 2 ( R + ), B = C ; L 0 y = − y ′′ + q ( x ) y , loc ([0 , ∞ )) | y (0) = y ′ (0) = 0 , − y ′′ + qy ∈ L 2 ( R + ) } , Dom L 0 = { y ∈ L 2 ( R + ) ∩ H 2 the potential q is such that L 0 � γ 1 ; Γ 2 y = y ′ (0) − y (0) φ ′ (0) Γ 1 y = − y (0) φ , φ , η ′ (0) where φ obeys − φ ′′ + qφ = 0, φ (0) = 1, φ ∈ L 2 ( R + ) and η = L − 1 φ .

2. Dynamical system with boundary control Recall that L 0 � γ 1 . The DSBC is u tt + L ∗ 0 u = 0 in H , t > 0 ; u | t =0 = u t | t =0 in H ; Γ 1 u = f ( t ) , t � 0; f is a boundary control of the ‘smooth’ class M := { f ∈ C ∞ ([0 , ∞ ); B ) | f ≡ 0 near t = 0 } , u = u f ( t ) is a solution ( wave ). For f ∈ M the DSBC has a unique classical solution. Reachable sets: U T := { u f ( T ) | f ∈ M } , U T ↑ as T ↑ . T > 0; Example: the Sturm–Liouville DSBC u tt − u xx + qu = 0 , x > 0 , t > 0; u | t =0 = u t | t =0 = 0 , x � 0; u | x =0 = f ( t ) , t � 0; u = u f ( x, t ) is the wave. Here U T = L 2 (0 , T ) (the speed of waves equals 1!). DSBC is determined by L 0 !

3. Dual system Recall that L = L ∗ is the Friedrichs extension of L 0 . The system dual to DSBC is v tt + Lv = h in H , t > 0; v | t =0 = v t | t =0 = 0 in H . For h ∈ C ∞ ([0 , ∞ ); H ), h ≡ 0 near t = 0 there a unique classical solution v = v h ( t ). Reachable sets: for A ∈ L [ H ] define V T A := { v h ( T ) | h ∈ C ∞ ([0 , ∞ ); A ) , h ≡ 0 near t = 0 } . Note that V T A ↑ as T ↑ . Define I T : L [ H ] → L [ H ] , I T A := V T A Monotonicity: if A ⊂ B , T 1 � T 2 then I T 1 A ⊂ I T 2 B . Isotony: I := { I T | T � 0 } , I 0 := id . Isotony is determined by L 0 !

Example: the system dual to Sturm–Liouville DSBC v tt − v xx + qv = h, x > 0 , t > 0; v | t =0 = v t | t =0 = 0 , x � 0; v = v h ( x, t ) is the solution. Lemma. The S.–L. isotony acts as I T L 2 ( a, b ) = L 2 (max { 0 , a − T } , b + T ) , T > 0 . ( ⋆ ) Hilbert lattice of subspaces L [ H ] (a reminder): F , G ∈ L [ H ] implies Operations: F ∧ G := F ∩ G , F ∨ G := span { F , G } , F ⊥ := H ⊖ F ∈ L [ H ] . Topology: F j → F ⇔ s − lim j →∞ P j = P , where P j , P are the orthogonal projections on F j , F .

Boundary lattice of DSBC: the minimal (sub)lattice L U ⊂ L [ H ] that obeys U T ∈ L U I T L U ⊂ L U , and T > 0 . Boundary lattice is determined by L 0 ! Example: the boundary lattice of Sturm–Liouville DSBC is � � � � n � � L U = G ∈ L [ H ] � G = L 2 ( a j , b j ) , 0 � a 1 < b 1 < a 2 < b 2 < · · · < a n < b n � ∞ j =1 (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 ⊥ )( t ) := ( f ( t )) ⊥ , ( f ∧ g )( t ) := f ( t ) ∧ g ( t ) , ( f ∨ g )( t ) := f ( t ) ∨ g ( t ) , t � 0 L [ H ] and the pointwise convergence f j → f ⇔ f j ( t ) → f ( t ) , t � 0. Define F U := { f ∈ F | f ( t ) = I t [ f (0)] , t � 0 , f (0) ∈ L U } ⊂ 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 L 0 : F U is a partially ordered set with the order ⊆ . The wave spectrum of L 0 is Ω L 0 := At F U (the closure is important!). Distance on Ω L 0 : d ( α, β ) = 2 inf { t > 0 | α ( t ) ∧ β ( t ) � = 0 } (interaction time). Wave spectrum is a unitary invariant of L 0 !

Wave model: Transform U : elements y ∈ H �→ functions � y = Uy on the wave spectrum by � � P α ( t ) y, e H � � y ( α ) := lim � , α ∈ Ω L 0 , P α ( t ) e, e t → +0 H where α = α ( t ) is an atom (an H -valued function of time), P α ( 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 L 0 given in any representation (i.e., by any unitary copy of L 0 ). Available for inverse problems:  charact. function of L 0 ;       spectral data; Determines L 0       Weyl function of the up to unitary    Inverse Wave   ⇒ Green system; ⇒ equivalence ⇒   Data  model    scattering data; ( L 0 is completely     boundary spectral non-selfadjoint)    and dynamical data

5. Examples 1. H = L 2 ( R + ) ( R + = { x � 0 } ), B = C ; the Sturm–Liouville operator L 0 = − d 2 dx 2 + q ( x ) on � � loc ( R + ) | y (0) = y ′ (0) = 0 , − y ′′ + qy ∈ L 2 ( R + ) y ∈ L 2 ( R + ) ∩ H 2 Dom L 0 = , q is such that L 0 � γ 1 ( γ > 0); loc ( R + ) | y (0) = 0 , − y ′′ + qy ∈ L 2 ( R + ) } , Dom L = { y ∈ L 2 ( R + ) ∩ H 2 loc ( R + ) | − y ′′ + qy ∈ L 2 ( R + ) } : L 0 ⊂ L ⊂ L ∗ 0 = { y ∈ L 2 ( R + ) ∩ H 2 Dom L ∗ 0 . � Wave Model: H = L 2 ( R + ) ( R + = { τ � 0 } ), B = C ; L 0 = − κ − 1 d 2 � dτ 2 κ + 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 = L 2 (Ω), B = L 2 ( ∂ Ω); the Beltrami–Laplace operator L 0 = − ∆ on Dom L 0 = { y ∈ H 2 (Ω) | y | ∂ Ω = ∂ ν y | ∂ Ω = 0 } ( ν is the outward normal). isometric ∼ Wave spectrum: Ω L 0 Ω. = � L 0 = κ − 1 L 0 κ in L 2 (Ω L 0 ) ( κ is a smooth positive function). Wave model: Inverse Problem: given the boundary inverse data to recover Ω. Comparing the models  Sz.-Nagy–Foias  operator of multiplication by independent variable Pavlov in a model Hilbert space of functions  Strauss 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.