Inverse Problem for Quantum Graphs: Magnetic Boundary Control Pavel - PowerPoint PPT Presentation

Inverse Problem for Quantum Graphs: Magnetic Boundary Control Pavel Kurasov December 21, 2019 Vienna Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 1 / 18

Quantum graph Metric graph ✄✄❍❍❍❍❍❍❍❍ ❆ ♣ ❆ ✄ ❆❜❜❜ ✄ � ♣ ❍ ✥ ❜✥✥✥ ✄ � � ♣ ♣ ♣ ✄ � � ♣ P P P ♣ � P � � P P P � ♣ ♣ Differential expression on the edges � � 2 i d ℓ q , a = dx + a ( x ) + q ( x ) Matching conditions Via irreducible unitary matrices S m associated with each internal vertex V m i ( S m − I ) � ψ m = ( S m + I ) ∂ � ψ m , m = 1 , 2 , . . . , M . Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 2 / 18

Inverse problem Task : reconstruct all three members from the family: the metric graph the potential(s) q ( x ) and a ( x ); the vertex conditions. Contact set ∂ Γ - a fixed set of vertices containing all degree one vertices. Boundary control - solution to the wave equation subject to control conditions on ∂ Γ  � 2 u ( x , t ) + q ( x ) u ( x , t ) = − ∂ 2 � i d dx + a ( x ) ∂ t 2 u ( x , t )    u ( x , 0) = u t ( x , 0) = 0 ,   u ( · , t ) | ∂ Γ = �  f ( t ) Response operator R T : � f ( t ) �→ ∂� u ( · , t ) | ∂ Γ . ���� = � u ( · , t ) | ∂ Γ Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 3 / 18

Response operator and M -function Response operator R T : � f ( t ) �→ ∂� u ( · , t ) | ∂ Γ . ���� = � u ( · , t ) | ∂ Γ Titchmarsh-Weyl matrix-valued M -function : Consider ψ ( x , λ ) - solution to the stationary equation � � 2 i d dx + a ( x ) ψ ( x , λ ) + q ( x ) ψ ( x , λ ) = λψ ( x , λ ) M ( λ ) : � ψ | ∂ Γ �→ ∂ � ψ | ∂ Γ Connection � � � ( s ) = M ( − s 2 )ˆ R � � f f ( s ) where ˆ · denotes the Laplace transform. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 4 / 18

Two explicit formulas � ∞ � − 1 � ψ st n | ∂ Γ , ·� ℓ 2 ( ∂ Γ) ψ st n | ∂ Γ � M Γ ( λ ) = − , λ st n − λ n =1 where λ st n and ψ st n are the eigenvalues and ortho-normalised eigenfunctions of L st . ∞ λ − λ ′ � M Γ ( λ ) − M Γ ( λ ′ ) = n − λ ′ ) � ∂ψ D n | ∂ Γ , ·� C B ∂ψ D n | ∂ Γ , ( λ D n − λ )( λ D n =1 where λ D n and ψ D n are the eigenvalues and eigenfunctions of the Dirichlet odinger operator L D . Schr¨ These formulas indicate where zeroes and singularities of the M -functions may be situated. Existence of invisible eigenfunctions. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 5 / 18

Two explicit formulas � ∞ � − 1 � ψ st n | ∂ Γ , ·� ℓ 2 ( ∂ Γ) ψ st n | ∂ Γ � M Γ ( λ ) = − , λ st n − λ n =1 where λ st n and ψ st n are the eigenvalues and ortho-normalised eigenfunctions of L st . ∞ λ − λ ′ � M Γ ( λ ) − M Γ ( λ ′ ) = n − λ ′ ) � ∂ψ D n | ∂ Γ , ·� C B ∂ψ D n | ∂ Γ , ( λ D n − λ )( λ D n =1 where λ D n and ψ D n are the eigenvalues and eigenfunctions of the Dirichlet odinger operator L D . Schr¨ These formulas indicate where zeroes and singularities of the M -functions may be situated. Existence of invisible eigenfunctions. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 5 / 18

Limitations The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) � Φ j = a ( y ) dy ⇒ fluxes Φ j C j Vertex conditions can be determined up to phases corresponding to internal edges: ψ ( x ) = e i θ n ˆ ψ ( x ) , x ∈ E n . ⇒ phases Θ n One has to require that reflection and transmission from the vertices is non-trivial. Ex . x 3   1 1 0 0 √ √ 2 2 1 1 x 1 x 2  0 0  √ √ x 4 S 1 =   2 2 x 8  1 − 1  x 6 x 5 0 0 √ √   2 2 1 − 1 0 0 √ √ 2 2 � � S m ( ∞ ) ij � = 0 x 7 Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

Limitations The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) � Φ j = a ( y ) dy ⇒ fluxes Φ j C j Vertex conditions can be determined up to phases corresponding to internal edges: ψ ( x ) = e i θ n ˆ ψ ( x ) , x ∈ E n . ⇒ phases Θ n One has to require that reflection and transmission from the vertices is non-trivial. Ex . x 3   1 1 0 0 √ √ 2 2 1 1 x 1 x 2  0 0  √ √ x 4 S 1 =   2 2 x 8  1 − 1  x 6 x 5 0 0 √ √   2 2 1 − 1 0 0 √ √ 2 2 � � S m ( ∞ ) ij � = 0 x 7 Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

Limitations The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) � Φ j = a ( y ) dy ⇒ fluxes Φ j C j Vertex conditions can be determined up to phases corresponding to internal edges: ψ ( x ) = e i θ n ˆ ψ ( x ) , x ∈ E n . ⇒ phases Θ n One has to require that reflection and transmission from the vertices is non-trivial. Ex . x 3   1 1 0 0 √ √ 2 2 1 1 x 1 x 2  0 0  √ √ x 4 S 1 =   2 2 x 8  1 − 1  x 6 x 5 0 0 √ √   2 2 1 − 1 0 0 √ √ 2 2 � � S m ( ∞ ) ij � = 0 x 7 Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

Inverse problems for trees Geometric perturbations peeling leafs; trimming brunches; cutting brunches. Cleaning procedure removing potential q on the boundary edges Schr¨ odinger ⇒ Laplace All four principles are based on two Lemmas concerning gluing extensions of symmetric operators. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 7 / 18

Gluing graphs ⇒ Two graphs with 4 and 3 contact vertices are transformed into a graph with 3 contact vertices. One gluing point: ⇒ Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 8 / 18

Gluing extensions of symmetric operators Let A 1 and A 2 be two symmetric operators with the boundary values: � A ∗ j u j , v j � − � u j , A ∗ u ∂ u ∂ v ∂ j v j � = � � j , ∂� v j � − � ∂� j , � j � . M -functions ψ ∂ = ∂� M 1 , 2 ( λ ) � u ∂ , where A ∗ j ψ j = λψ j . Consider the symmetric extension A = A 1 ⊕ A 2 determined by the coupling conditions: u ∂ u ∂ u ∂ u ∂ � 1 | L 1 = � 2 | L 2 ; ∂� 1 | L 1 = − ∂� 2 | L 2 , where L 1 , L 2 are two identified subspaces of the same dimension. Lemma 1 (following Schur-Frobenius) The M -functions associated with the operators A 1 , A 2 , and A are related via � M 22 � 1 − M 21 1 ( M 11 1 + M 11 2 ) − 1 M 12 − M 21 1 ( M 11 1 + M 11 2 ) − 1 M 12 1 2 M ( λ ) = . − M 21 2 ( M 11 1 + M 11 2 ) − 1 M 12 M 22 2 − M 21 2 ( M 11 1 + M 11 2 ) − 1 M 12 1 2 where M lm come from the block decomposition of M Γ j . j Lemma 2 Assume that dim L 1 ≡ dim L 2 = 1 , then any two out of three M -functions in the above formula determine uniquely the third one. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 9 / 18

Gluing extensions of symmetric operators Let A 1 and A 2 be two symmetric operators with the boundary values: � A ∗ j u j , v j � − � u j , A ∗ u ∂ u ∂ v ∂ j v j � = � � j , ∂� v j � − � ∂� j , � j � . M -functions ψ ∂ = ∂� M 1 , 2 ( λ ) � u ∂ , where A ∗ j ψ j = λψ j . Consider the symmetric extension A = A 1 ⊕ A 2 determined by the coupling conditions: u ∂ u ∂ u ∂ u ∂ � 1 | L 1 = � 2 | L 2 ; ∂� 1 | L 1 = − ∂� 2 | L 2 , where L 1 , L 2 are two identified subspaces of the same dimension. Lemma 1 (following Schur-Frobenius) The M -functions associated with the operators A 1 , A 2 , and A are related via � M 22 � 1 − M 21 1 ( M 11 1 + M 11 2 ) − 1 M 12 − M 21 1 ( M 11 1 + M 11 2 ) − 1 M 12 1 2 M ( λ ) = . − M 21 2 ( M 11 1 + M 11 2 ) − 1 M 12 M 22 2 − M 21 2 ( M 11 1 + M 11 2 ) − 1 M 12 1 2 where M lm come from the block decomposition of M Γ j . j Lemma 2 Assume that dim L 1 ≡ dim L 2 = 1 , then any two out of three M -functions in the above formula determine uniquely the third one. Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 9 / 18