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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

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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

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SLIDE 2

Quantum graph

Metric graph

  • P

P P P P P P ✄ ✄ ✄ ✄ ✄✄❍❍❍❍❍❍❍❍ ❍

  • ❆

❆ ❆❜❜❜ ❜✥✥✥ ✥

  • ♣

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ Differential expression on the edges ℓq,a =

  • i d

dx + a(x) 2 + q(x) Matching conditions Via irreducible unitary matrices Sm associated with each internal vertex Vm i(Sm − I) ψm = (Sm + I)∂ ψm, m = 1, 2, . . . , M.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 2 / 18

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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

  • n ∂Γ

      

  • i d

dx + a(x)

2 u(x, t) + q(x)u(x, t) = − ∂2

∂t2 u(x, t)

u(x, 0) = ut(x, 0) = 0, u(·, t)|∂Γ = f (t) Response operator RT :

  • f (t)
  • =

u(·,t)|∂Γ

→ ∂ u(·, t)|∂Γ.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 3 / 18

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SLIDE 4

Response operator and M-function

Response operator RT :

  • f (t)
  • =

u(·,t)|∂Γ

→ ∂ u(·, t)|∂Γ. Titchmarsh-Weyl matrix-valued M-function: Consider ψ(x, λ) - solution to the stationary equation

  • i d

dx + a(x) 2 ψ(x, λ) + q(x)ψ(x, λ) = λψ(x, λ) M(λ) : ψ|∂Γ → ∂ ψ|∂Γ Connection

  • R

f

  • (s) = M(−s2)ˆ
  • f (s)

where ˆ · denotes the Laplace transform.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 4 / 18

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SLIDE 5

Two explicit formulas

MΓ(λ) = − ∞

  • n=1

ψst

n |∂Γ, ·ℓ2(∂Γ)ψst n |∂Γ

λst

n − λ

−1 , where λst

n and ψst n are the eigenvalues and ortho-normalised eigenfunctions of

Lst. MΓ(λ) − MΓ(λ′) =

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)∂ψD n |∂Γ, ·CB∂ψD n |∂Γ,

where λD

n and ψD n are the eigenvalues and eigenfunctions of the Dirichlet

Schr¨

  • dinger operator LD.

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

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Two explicit formulas

MΓ(λ) = − ∞

  • n=1

ψst

n |∂Γ, ·ℓ2(∂Γ)ψst n |∂Γ

λst

n − λ

−1 , where λst

n and ψst n are the eigenvalues and ortho-normalised eigenfunctions of

Lst. MΓ(λ) − MΓ(λ′) =

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)∂ψD n |∂Γ, ·CB∂ψD n |∂Γ,

where λD

n and ψD n are the eigenvalues and eigenfunctions of the Dirichlet

Schr¨

  • dinger operator LD.

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

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SLIDE 7

Limitations

The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) Φj =

  • Cj

a(y)dy ⇒ fluxes Φj Vertex conditions can be determined up to phases corresponding to internal edges: ψ(x) = eiθn ˆ ψ(x), x ∈ En. ⇒ phases Θn One has to require that reflection and transmission from the vertices is non-trivial. Ex. x2 x1 x3 x4 x5 x6 x7 x8 S1 =     

1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2

− 1

√ 2 1 √ 2

− 1

√ 2

    

  • Sm(∞)
  • ij = 0

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

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Limitations

The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) Φj =

  • Cj

a(y)dy ⇒ fluxes Φj Vertex conditions can be determined up to phases corresponding to internal edges: ψ(x) = eiθn ˆ ψ(x), x ∈ En. ⇒ phases Θn One has to require that reflection and transmission from the vertices is non-trivial. Ex. x2 x1 x3 x4 x5 x6 x7 x8 S1 =     

1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2

− 1

√ 2 1 √ 2

− 1

√ 2

    

  • Sm(∞)
  • ij = 0

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

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Limitations

The exact form of the magnetic potential plays no role ⇒ magnetic fluxes through cycles (if any) Φj =

  • Cj

a(y)dy ⇒ fluxes Φj Vertex conditions can be determined up to phases corresponding to internal edges: ψ(x) = eiθn ˆ ψ(x), x ∈ En. ⇒ phases Θn One has to require that reflection and transmission from the vertices is non-trivial. Ex. x2 x1 x3 x4 x5 x6 x7 x8 S1 =     

1 √ 2 1 √ 2 1 √ 2 1 √ 2 1 √ 2

− 1

√ 2 1 √ 2

− 1

√ 2

    

  • Sm(∞)
  • ij = 0

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 6 / 18

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SLIDE 10

Inverse problems for trees

Geometric perturbations peeling leafs; trimming brunches; cutting brunches. Cleaning procedure removing potential q on the boundary edges Schr¨

  • dinger

⇒ 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

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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

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Gluing extensions of symmetric operators

Let A1 and A2 be two symmetric operators with the boundary values: A∗

j uj, vj − uj, A∗ j vj =

u∂

j , ∂

vj − ∂ u∂

j ,

v ∂

j .

M-functions M1,2(λ) ψ∂ = ∂ u∂, where A∗

j ψj = λψj.

Consider the symmetric extension A = A1 ⊕ A2 determined by the coupling conditions:

  • u∂

1 |L1 =

u∂

2 |L2;

∂ u∂

1 |L1 = −∂

u∂

2 |L2,

where L1, L2 are two identified subspaces of the same dimension. Lemma 1 (following Schur-Frobenius) The M-functions associated with the operators A1, A2, and A are related via M(λ) = M22

1 − M21 1 (M11 1 + M11 2 )−1M12 1

−M21

1 (M11 1 + M11 2 )−1M12 2

−M21

2 (M11 1 + M11 2 )−1M12 1

M22

2 − M21 2 (M11 1 + M11 2 )−1M12 2

  • .

where Mlm

j

come from the block decomposition of MΓj. Lemma 2 Assume that dim L1 ≡ dim L2 = 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

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Gluing extensions of symmetric operators

Let A1 and A2 be two symmetric operators with the boundary values: A∗

j uj, vj − uj, A∗ j vj =

u∂

j , ∂

vj − ∂ u∂

j ,

v ∂

j .

M-functions M1,2(λ) ψ∂ = ∂ u∂, where A∗

j ψj = λψj.

Consider the symmetric extension A = A1 ⊕ A2 determined by the coupling conditions:

  • u∂

1 |L1 =

u∂

2 |L2;

∂ u∂

1 |L1 = −∂

u∂

2 |L2,

where L1, L2 are two identified subspaces of the same dimension. Lemma 1 (following Schur-Frobenius) The M-functions associated with the operators A1, A2, and A are related via M(λ) = M22

1 − M21 1 (M11 1 + M11 2 )−1M12 1

−M21

1 (M11 1 + M11 2 )−1M12 2

−M21

2 (M11 1 + M11 2 )−1M12 1

M22

2 − M21 2 (M11 1 + M11 2 )−1M12 2

  • .

where Mlm

j

come from the block decomposition of MΓj. Lemma 2 Assume that dim L1 ≡ dim L2 = 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

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SLIDE 14

Gluing extensions of symmetric operators

Let A1 and A2 be two symmetric operators with the boundary values: A∗

j uj, vj − uj, A∗ j vj =

u∂

j , ∂

vj − ∂ u∂

j ,

v ∂

j .

M-functions M1,2(λ) ψ∂ = ∂ u∂, where A∗

j ψj = λψj.

Consider the symmetric extension A = A1 ⊕ A2 determined by the coupling conditions:

  • u∂

1 |L1 =

u∂

2 |L2;

∂ u∂

1 |L1 = −∂

u∂

2 |L2,

where L1, L2 are two identified subspaces of the same dimension. Lemma 1 (following Schur-Frobenius) The M-functions associated with the operators A1, A2, and A are related via M(λ) = M22

1 − M21 1 (M11 1 + M11 2 )−1M12 1

−M21

1 (M11 1 + M11 2 )−1M12 2

−M21

2 (M11 1 + M11 2 )−1M12 1

M22

2 − M21 2 (M11 1 + M11 2 )−1M12 2

  • .

where Mlm

j

come from the block decomposition of MΓj. Lemma 2 Assume that dim L1 ≡ dim L2 = 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

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Three inverse problems

I Reconstruction of the metric graph Globally: Every metric tree is uniquely reconstructed from the travelling times between the boundary vertices Locally: Travelling times associated with a bunch determine the bunch II Reconstruction of the potential Boundary Control: (the diagonal of) the response operator RT determines the potential on the boundary edges everywhere at the distance d < T/2 from the boundary III Reconstruction of the vertex conditions The response operator for an equilateral bunch with zero potential determines the vertex conditions up to the phase θn associated with the root edge.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 10 / 18

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Three inverse problems

III Reconstruction of the vertex conditions The response operator for an equilateral bunch with zero potential determines the vertex conditions up to the phase θn associated with the root edge. The kernel of the response operator is of the form: r(t) = −δ′(t)+2PSv(∞)Pδ′(t−2ℓ)+4P(I−P−1)A(I−P−1)Pδ(t)+smooth where

◮ Sv(∞) - the limit of the vertex scattering matrix; ◮ P - the projector on the d − 1 components associated with the bunch; ◮ A - the matrix in the quadratic form parameterisation of vertex conditions; ◮ P−1 - the eigenprojector for S corresponding to e.v. −1.

Sv(∞) = I − 2P−1

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 11 / 18

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SLIDE 17

Fusion

1

Selecting a bunch in the tree. Using global or local procedure (Subproblem I)

2

Reconstruction of the potential on the bunch. Boundary Control determines potential on the boundary edges from the bunch (Subproblem II).

3

Cleaning of the bunch - removing potential. Based on gluing Lemmas.

4

Trimming of the bunch. Based on gluing Lemmas.

5

Recovering vertex conditions at the root. The response operator for equilateral bunch with zero potential determines the vertex condition at the root of the bunch (Subproblem III).

6

Cutting away bunches - pruning of the tree. Based on gluing Lemmas. Repeating this procedure sufficiently many times we solve the inverse problem for trees, of course up to the phases θn at internal edges. Assumptions: q ∈ L1(Γ), Sv(∞) does not have zero entries.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 12 / 18

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Inverse problems for graphs with cycles

Magnetic potential is equivalent to change of vertex conditions ⇒ only standard vertex conditions (continuity + Kirchhoff). The response operator and M-matrices are considered as functions of the set of fluxes Φj, fixed each time. Magnetic boundary control M(λ, Φj) → q(x) Key feature: existence of invisible eigenfunctions Every such eigenfunction produces ambiguity in the solution of the inverse problem.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 13 / 18

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SLIDE 19

Loop graph

V1

x1 x2 x3

MΓloop

ℓ1

(λ, Φ) = M11 + 2 cos ΦM12 + M22 = 2 cos Φ − Tr T(k) t12(k) , where T(k) = {tij(k)} is the transfer matrix for [x1, x2]. NB! The M-function depends on cos Φ, i.e. Φ and −Φ lead to the same MΓloop

ℓ1 Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 14 / 18

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Solution using the transfer matrix

Solving inverse problem: t12(k) = 1 M12(λ); u+(k) := 1 2

  • t11(k) + t22(k)
  • =

1 2 Tr M(λ) M12(λ) . t12(kj) = 0 - the spectrum of the Dirichlet-Diricihlet operator on [x1, x2]. At these points we know: t11(kj)t22(kj) = 1 det T(k) = 1; t11(kj) + t22(kj) = u+(kj) − known function. ⇒ t11(kj), t22(kj) are determined up to a sign tii(k) are analytic functions of exponential type and are uniquely determined by their values at kj, provided

  • q(x)dx = 0.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 15 / 18

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Solution using the M-function

M12(λ) = 1 4

  • MΓloop

ℓ1

(λ, 0) − MΓloop

ℓ1

(λ, π)

  • ,

Tr M(λ) = 1 2

  • MΓloop

ℓ1

(λ, 0) + MΓloop

ℓ1

(λ, π)

  • .

Consider λD

j = k2 j and check for which (complex) Φ the M-function does not

have a singularity at this point. MΓ(λ, Φ)) = MΓ(λ′, Φ)+

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)∂ψD n (Φ)|∂Γ, ·ℓ2(∂Γ)∂ψD n (Φ)|∂Γ,

Find Φj for which the M-function is not singular at λD

j . NB! The phase Φj is

determined up to a sign, since the formula for M(λ, Φ) contains cos Φ. Invisible eigenfunction ψ(x1) = ψ(x2) = 0, ψ′(x1) − eiΦjψ′(x2) = 0. ⇒ ψ′(x1) ψ′(x2) = eiΦj

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 16 / 18

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SLIDE 22

Three scalar M-functions

  • M(λ)
  • 11

=

  • MΓ(λ′)
  • 11 +

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)|∂ψD n (x1)|2

  • M(λ)
  • 22

=

  • MΓ(λ′)
  • 22 +

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)|∂ψD n (x2)|2

Tr M(λ) = Tr M(λ′) +

∞

  • n=1

λ − λ′ (λD

n − λ)(λD n − λ′)

  • |∂ψD

n (x1)|2 + |∂ψD n (x2)|2

ψ′(x1) ψ′(x2)

  • = eiΦj

|∂ψD

n (x1)|2 + |∂ψD n (x2)|2

   ⇒ |∂ψD

n (x1)|2, |∂ψD n (x2)|2

Conclusion: Tr M(λ) determines

  • M(λ)
  • 11,
  • M(λ)
  • 22

up to a real constant.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 17 / 18

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SLIDE 23

Cycle opening procedure

Note that the last formula does not require that M is an M-function for an interval - we did not use properties of the transfer matrix. ⇒ ⇒ ⇒ Each cycle requires a sequence of signs + a real constant.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 18 / 18

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Cycle opening procedure

Note that the last formula does not require that M is an M-function for an interval - we did not use properties of the transfer matrix. ⇒ ⇒ ⇒ Each cycle requires a sequence of signs + a real constant.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 18 / 18

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Cycle opening procedure

Note that the last formula does not require that M is an M-function for an interval - we did not use properties of the transfer matrix. ⇒ ⇒ ⇒ Each cycle requires a sequence of signs + a real constant.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 18 / 18

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Cycle opening procedure

Note that the last formula does not require that M is an M-function for an interval - we did not use properties of the transfer matrix. ⇒ ⇒ ⇒ Each cycle requires a sequence of signs + a real constant.

Kurasov (Stockholm) Magnetic Boundary Control December 21, 2019, Vienna 18 / 18