Wave model of the Schroedinger operator on semiaxis (the limit point - PDF document

Wave model of the Schroedinger operator on semiaxis (the limit point case) M.I.Belishev, S.A.Simonov (PDMI) Plan 1. Operator L 0 • The class of operators. Schroedinger operator L Sch on (0 , ∞ ) with defect 0 indexes (1,1). • A Green System. The Green system of L Sch 0 . 2. Dynamical System with Boundary Control • An abstract DSBC associated with L 0 . Reachable sets, controllability. • Isotony I L 0 . The system α L Sch and isotony I L Sch 0 . 0 3. Wave model • Lattices, lattice-valued functions, atoms. A space Ω L 0 (wave spectrum of L 0 ). The space Ω L Sch 0 . • A wave model ˜ 0 . The model ˜ L ∗ L Sch ∗ . 0 • Applications to inverse problems. The spectral IP for L Sch 0 . Operator L 0 1 Let H be a separable Gilbert space, L 0 acts in H and 1. L 0 = ¯ L 0 , Dom L 0 = H 2. ∃ κ = const > 0 s.t. ( L 0 y, y ) � κ � y � 2 , y ∈ Dom L 0 1

2 3. 1 � n ± ( L 0 ) = dim Ker L ∗ 0 � ∞ . Let L be the Friedichs extension of L 0 : L ∗ = L ; ( Ly, y ) � κ � y � 2 , y ∈ Dom L . L 0 ⊂ L ⊂ L ∗ 0 ; ========================================= Notation: R + := (0 , ∞ ) , ¯ R + := [0 , ∞ ). Let H = L 2 ( R + ) and L 0 = L Sch 0 , 0 y := − y ′′ + qy , L Sch � � R + ) | y (0) = y ′ (0) = 0; − y ′′ + qy ∈ H loc (¯ Dom L Sch y ∈ H ∩ H 2 = , 0 where q = q ( x ) is a real-valued function ( potential ) provided � ¯ � R + (1) q ∈ C loc n ± ( L Sch (2) 0 ) = 1 (the limit point case) 0 y, y ) � κ � y � 2 , y ∈ Dom L Sch (3) ( L Sch . 0 In such a case, the problem − φ ′′ + qφ = 0 , x > 0 ; φ ∈ L 2 ( R + ) φ (0) = 1 , has a unique solution φ ( x ); 0 ) ∗ y := − y ′′ + qy , ( L Sch � � 0 ) ∗ = R + ) | − y ′′ + qy ∈ H loc (¯ Dom ( L Sch y ∈ H ∩ H 2 , 0 ) ∗ = { cφ | c ∈ C } ; Ker ( L Sch the Friedrichs extension of L Sch is 0 L Sch y := − y ′′ + qy , � � Dom L Sch = R + ) | y (0) = 0; − y ′′ + qy ∈ H loc (¯ y ∈ H ∩ H 2 . =========================================

3 Green system Let H B be the Hilbert spaces, A : H → H and Γ i : H → B ( i = 1 , 2) the operators provided: Dom A = H , Dom Γ i ⊃ Dom A, Ran Γ 1 ∨ Ran Γ 2 = B . The collection G = {H , B ; A, Γ 1 , Γ 2 } is a Green system if ( Au, v ) H − ( u, Av ) H = (Γ 1 u, Γ 2 v ) B − (Γ 2 u, Γ 1 v ) B for u, v ∈ Dom A . H is an inner space , B is a boundary values space , A is a basic operator , Γ 1 , 2 are the boundary operators . System G L 0 Operator L 0 determines a Green system in a canonical way. Put K := Ker L ∗ Γ 1 := L − 1 L ∗ 0 − I , Γ 2 := P K L ∗ 0 , 0 , where P K projects in H onto K Proposition 1. The collection G L 0 := {H , K ; L ∗ 0 , Γ 1 , Γ 2 } is a Green system. ========================================= Let L 0 = L Sch 0 , H = L 2 ( R + ), K = { cφ | c ∈ C } ; � y ′ (0) − y (0) φ ′ (0) � Γ 1 y = − y (0) φ , Γ 2 y = φ , η ′ (0) where η := ( L Sch ) − 1 φ . Then G L Sch := {H , K ; ( L Sch 0 ) ∗ , Γ 1 , Γ 2 } is the canonical 0 Green system associated with L Sch 0 . ========================================= 2 Dynamical System with Boundary Control System α L 0 The Green system G L 0 determines a DSBC α L 0 of the form u tt + L ∗ 0 u = 0 , t > 0 , u | t =0 = u t | t =0 = 0 in H Γ 1 u = h , t � 0 ,

4 where h = h ( t ) is a boundary control (a K -valued function of the time), u = u h ( t ) is a solution ( wave , an H -valued function of the time). For a smooth enough class M ∋ h , u h is classical. Proposition 2. For an h ∈ M , one has � t � � L − 1 1 u h ( t ) = − h ( t ) + 2 sin ( t − s ) L h tt ( s ) ds , t � 0 , (2.1) 2 0 where L is the Friedrichs extension of L 0 . ========================================= For L 0 = L Sch 0 , system α L Sch is 0 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 � 0 . For M := { f ∈ C ∞ [0 , ∞ ) | supp f ⊂ (0 , ∞ ) } , the solution u f is classical. ========================================= Controllability For the DSBC α L 0 , the set U T L 0 := { u h ( T ) | h ∈ M} is called reachable (at the moment t = T ); � U T U L 0 := L 0 T> 0 is a total reachable set. System α L 0 is controllable , if U L 0 = H . Proposition 3. System α L 0 is controllable iff L 0 is a completely nonself- adjoint operator (i.e., L 0 induces a self-adjoint part in no subspace of H ). ========================================= For L 0 = L Sch 0 , one has = { y ∈ C ∞ (¯ U T R + ) | supp y ⊂ [0 , T ] } , = L 2 ( R + ) = H , U L Sch L Sch 0 0 so that α L Sch is controllable. 0 =========================================

5 Isotony Introduce a dynamical system β L 0 : v tt + Lv = g , t > 0 , v | t =0 = v t | t =0 = 0 , where g = g ( t ) is an H -valued function. For a smooth enough g , the solution v = v g ( t ) is � t � � L − 1 1 v g ( t ) = 2 sin ( t − s ) L g ( s ) ds , t � 0 2 0 holds. Fix a subspace G ⊂ H ; let g be a G -valued function. A set � � v g ( t ) | g ∈ L loc V t G := 2 ([0 , ∞ ); G ) is a reachable set of system β L 0 . Proposition 4. If G ⊂ G ′ and t � t ′ then V t G ⊂ V t ′ G ′ . Let L ( H ) be the lattice of subspaces of H with the standard operations A ∨ B = { a + b | a ∈ A , b ∈ B} , A ∧ B = A ∩ B , A ⊥ = H ⊖ A , the partial order ⊆ , the least and greatest elements { 0 } and H , and the relevant topology. A family of maps I = { I t } t � 0 , I t : L ( H ) → L ( H ) is said to be an isotony if I 0 := id I t G ⊂ I t ′ G ′ , G ⊂ G ′ , t � t ′ and implies i.e., I respects the natural order on L ( H ) × [0 , T ). By Proposition 4, the family I 0 I t L 0 := id ; L 0 G := V t t > 0 G , is the isotony determined by L 0 . ========================================= For E ⊂ ¯ R + , let E t := { x ∈ ¯ R + | dist ( x, E ) := inf e ∈ E | x − e | < t } , t > 0 (a metric neighborhood). Denote ∆ a,b := [ a, b ] ⊂ ¯ R + and L 2 (∆ a,b ) := { y ∈ L 2 ( R + ) | supp y ⊂ [ a, b ] } .

6 Proposition 5. For any 0 � a < b � ∞ , the relation I t 0 L 2 (∆ a,b ) = L 2 (∆ t a,b ) , t > 0 L Sch holds. ========================================= 3 Wave model Lattices Let L L 0 ⊂ L ( H ) be a minimal (sub)lattice s.t. T I t U L 0 ⊂ L L 0 , T � 0 and L 0 L L 0 ⊂ L L 0 , t � 0 (i.e., L L 0 is invariant w.r.t. the isotony I L 0 ). The space of the growing L L 0 -valued functions � � F ( · ) | F ( t ) = I t F I L 0 ([0 , ∞ ); L L 0 ) := L 0 A , t � 0 , A ∈ L L 0 is a lattice w.r.t. the point-wise operations, order, and topology. Wave spectrum Reminder. P is a partially ordered set, 0 is the least element. An a ∈ P is an atom ( minimal element ) if 0 � = p � a implies p = a . Let At P be the set of atoms in P . Basic definition. The set Ω L 0 := At F I L 0 ([0 , ∞ ); L L 0 ) endowed with a relevant topology is a wave spectrum of operator L 0 . Each ω = ω ( · ) ∈ Ω L 0 is a growing L L 0 -valued function of t . So, the path is L 0 ⇒ G L 0 ⇒ α L 0 , U T L 0 ⇒ I L 0 ⇒ L L 0 ⇒ F I L 0 ([0 , ∞ ); L L 0 ) ⇒ Ω L 0 .

7 ========================================= Theorem 1. For L 0 = L Sch 0 , there is a bijection ∋ ω ↔ x ∈ ¯ R + Ω L Sch 0 and each atom is of the form � { x } t � ω = ω x ( t ) = L 2 , t � 0 . Endowing the wave spectrum with the proper metrizable topology, one gets isom ¯ R + . Ω L Sch = 0 ========================================= The model Y Goal: the image map H ∋ y �→ ˜ y , where ˜ y = ˜ y ( · ) is a ”function” on Ω L 0 . Fix ω ∈ Ω L 0 ; let P t ω be the projection in H onto ω ( t ). For u, v ∈ H , a relation � � ω ∃ t 0 = t 0 ( ω, u, v ) s . t . P t ω u = P t = v ⇔ ω v for all t < t 0 u is an equivalence. • The equivalence class [ y ] ω is a germ of y ∈ H at the atom ω ∈ Ω L 0 . • The linear space G ω := { [ y ] ω | y ∈ H} is a germ space . • The space � ˜ H := G ω ω ∈ Ω L 0 is an image space . • The image map is Y : H → ˜ H , y ( ω ) := [ y ] ω , ˜ ω ∈ Ω L 0 . • The wave model of L ∗ 0 is 0 Y − 1 . � 0 : ˜ H → ˜ � 0 := Y L ∗ L ∗ L ∗ H ,

8 ========================================= 0 ) ∗ and fix an ω ∈ Ω L 0 . Let y ∈ H = L 2 ( R + ). Take an e = cφ ∈ Ker ( L Sch ≡ ¯ R + (by Theorem 1) and Identify Ω L Sch 0 ( P t ω y, e ) ω ∈ ¯ R + . y ( ω ) ≡ lim ˜ ω e, e ) , ( P t t → 0 Then ˜ H ≡ L 2 , µ ( R + ) with dµ = dω e 2 ( ω ) . Theorem 2. The representation � � � y ′′ ( ω ) + p ( ω ) ˜ y ′ ( ω ) + Q ( ω )˜ ( L Sch 0 ) ∗ ˜ ( ω ) = − ˜ y ( ω ) , ω > 0 y is valid with p := − 2 e ′ ( ω ) Q := q ( ω ) − e ′′ ( ω ) e ( ω ) , e ( ω ) . Application to IP Inverse Data: the spectral function, Weyl function, dynamical response operator , etc. Then 0 U ∗ ⇒ wave model � 0 ) ∗ ⇒ p, Q ⇒ q . ID ⇒ copy UL Sch ( L Sch ========================================= References [1] M.I.Belishev. A unitary invariant of a semi-bounded operator in recon- struction of manifolds. Journal of Operator Theory , Volume 69 (2013), Issue 2, 299-326. [2] M.I.Belishev and M.N.Demchenko. Dynamical system with boundary control associated with a symmetric semibounded operator. Journal of Mathematical Sciences , October 2013, Volume 194, Issue 1, pp 8-20. DOI: 10.1007/s10958-013-1501-8. [3] A.N.Kochubei. Extensions of symmetric operators and symmetric binary relations. Math. Notes , 17(1): 25–28, 1975. [4] V.Ryzhov. A General Boundary Value Problem and its Weyl Function. Opuscula Math. , 27 (2007), no. 2, 305–331.