On the Inverse Spectral Problem for Graphs with Cycles Pavel - - PowerPoint PPT Presentation

On the Inverse Spectral Problem for Graphs with Cycles

Pavel Kurasov

Lund University, SWEDEN

July 17, 2008

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 1 / 20

Introduction

1

Introduction Quantum graphs Spectral properties Inverse problems The main idea

2

Marchenko-Ostrovsky theory

3

Inverse problems for simple graphs Ring Lassoo Zweih¨ ander

4

General result

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 2 / 20

Introduction Quantum graphs

Quantum graph as a triplet

1

Metric graph Γ - union of intervals ∆j = [x2j−1, x2j] connected together at the vertices Vm considered as equivalence classes of end-points ⇒ the Hilbert space L2(Γ);

2

Differential expression (formally symmetric) on the edges Lq,a =

• −1

i d dx + a(x) 2 + q(x) ⇒ the linear operator Lq,a;

3

Boundary conditions at the vertices

to determine Lq,a as a self-adjoint operator, connect together different edges.

In this talk we are going to speak only about the standard boundary conditions only, that is: the function is continuous, the sum of ”normal” derivatives is zero

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 3 / 20

Introduction Quantum graphs

Quantum graph as a triplet

1

Metric graph Γ - union of intervals ∆j = [x2j−1, x2j] connected together at the vertices Vm considered as equivalence classes of end-points ⇒ the Hilbert space L2(Γ);

2

Differential expression (formally symmetric) on the edges Lq,a =

• −1

i d dx + a(x) 2 + q(x) ⇒ the linear operator Lq,a;

3

Boundary conditions at the vertices

to determine Lq,a as a self-adjoint operator, connect together different edges.

In this talk we are going to speak only about the standard boundary conditions only, that is: the function is continuous, the sum of ”normal” derivatives is zero

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 3 / 20

Introduction Spectral properties

”Elimination” of the magnetic field

Consider the unitary transformation: (Uψ)(x) = exp

• −i

x

x2n−1

a(y)dy

• ψ(x), x ∈ (x2n−1, x2n), n = 1, 2, ..., N,

which allows one to eliminate the magnetic field U

• (−1

i d dx + a(x))2 + q(x)

• U−1ψ(x) = − d2

dx2 ψ(x) + q(x)ψ(x). NB! The magnetic field can be eliminated from the differential expression, but then it appears in the boundary conditions (if the graph is not a tree). Proposition 1. The spectrum of the magnetic Schr¨

• dinger operator Lq,a is pure

discrete and does not depend on the particular form of the magnetic field but just on the fluxes of the magnetic field through the cycles Φj =

• cj

a(y)dy.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 4 / 20

Introduction Inverse problems

Inverse problems: concise historical overview

Solution of the inverse problem for quantum graphs means reconstruction

• f

the metric graph; the differential expressions on the edges; the coupling conditions at the vertices. Obtained results NB! for zero magnetic potential! Reconstruction of the graph:

with rationally independent lengths:

• B. Gutkin, T. Kottos and U. Smilansky, ’99, ’01;
• P. K., F. Stenberg and M. Nowaczyk ’02, ’05, ’07, ’08;

in the case of tree:

• V. Yurko ’06,
• S. Avdonin and P. K. ’08;

calculation of the Euler characteristic:

• P. K. ’08, ’08;

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 5 / 20

Introduction Inverse problems

Inverse problems: concise historical overview

Solution of the inverse problem for quantum graphs means reconstruction

• f

the metric graph; the differential expressions on the edges; the coupling conditions at the vertices. Obtained results NB! for zero magnetic potential! Reconstruction of the graph:

with rationally independent lengths:

• B. Gutkin, T. Kottos and U. Smilansky, ’99, ’01;
• P. K., F. Stenberg and M. Nowaczyk ’02, ’05, ’07, ’08;

in the case of tree:

• V. Yurko ’06,
• S. Avdonin and P. K. ’08;

calculation of the Euler characteristic:

• P. K. ’08, ’08;

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 5 / 20

Introduction Inverse problems

Reconstruction of the potential on graphs:

star graph: N.I. Gerasimenko and B.S. Pavlov, 1988; tree:

• M. Belishev and A. Vakulenko, ’04, ’06, ’07;
• M. Brown and R. Weikard, ’05;
• V. Yurko ’05, ’06, ’08;
• S. Avdonin and P. K. ’08;

impossibility for loops:

• J. Boman and P. K., 05
• V. Pivovarchik, manuscript;

Reconstruction of the boundary conditions:

for star graphs:

• V. Kostrykin and R. Schrader ’00, ’06;
• M. Harmer ’03.

Other references: Inverse problems for directed graphs: R. Carlson, ’99. General overview: P. Kuchment ’04.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 6 / 20

Introduction The main idea

The main idea

Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20

Introduction The main idea

The main idea

Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20

Introduction The main idea

The main idea

Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20

Introduction The main idea

The main idea

Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20

Introduction The main idea

The main idea

Conclusions concerning recovering the potential Knowldege of the spectrum alone is not enough to reconstruct the potential. Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) is an efficient tool to solve the inverse problem for graphs. Potential on the branches can be reconstructed from the TW function using Boundary Control method. Potential on the kernel of the graph in general cannot be determined by the TW function. Our programme Study the possibility to reconstruct the graph Γ and potential q on it from the TW function known for different values of the magnetic field.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 7 / 20

Marchenko-Ostrovsky theory

Marchenko-Ostrovsky theory

Provides necessary and sufficient conditions for a sequence of intervals to be the spectrum of one-dimensional periodic Schr¨

• dinger operator Lper

q .

Transfer matrix T(a, b; λ) − d2 dx2 ψ(x) + q(x)ψ(x) = λψ(x) ⇒ T(a, b; λ) : ψ(a) ψ′(a)

• →

ψ(b) ψ′(b)

• Introduce the functions:

u±(λ) = (t11(λ)±t22(λ))/2 The end points of the spectral intervals µj, ˜ µj are solutions to the equation u+(λ) = ±1.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 8 / 20

Marchenko-Ostrovsky theory

Proposition 2. For the sequences 0 = ˜ µ0 < µ1 ≤ ˜ µ1 < µ2 ≤ ˜ µ2 < . . . (1) to be the spectra of periodic and antiperiodic boundary value problems generated on the interval [0, π] by the operator − d2

dx2 + q(x) with real potential

q(x) ∈ W n

2 [0, π], it is necessary and sufficient that there exist a sequence of real

numbers hk (k = 0, ±1, ±2, . . . ) satisfying the conditions

∞

• k=1

(kn+1hk)2 < ∞, h0 = 0, hk = h−k ≥ 0(k = 1, 2, . . . ), (2) such that √µk = z(πk − 0),

• ˜

µk = z(πk + 0) (k = 1, 2, . . . ), where the function z(θ) effects a conformal mapping of the region {θ : Im θ > 0} \ ∪+∞

k=−∞ {θ : Re θ = kπ, 0 ≤ Im θ ≤ hk}

(3) into the upper half-plane, normalized by the conditions θ(0) = 0, lim

y→∞(iy)−1θ(iy) = π.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 9 / 20

Marchenko-Ostrovsky theory

In fact the following statement has been proven in V.A. Marchenko, I.V. Ostrovsky, A characterization of the spectrum of the Hill operator (Russian) Mat. Sb. (N.S.), 97 (139) (1975), no. 4(8), 540–606. Proposition 3. Assume that all conditions of Proposition 2 are satisfied. The following set of spectral data determine the potential uniquely: the spectrum of the periodic operator [0 = ˜ µ0, µ1] ∪ [˜ µ1, µ2] ∪ [˜ µ2, µ3] ∪ . . . , the D-D spectrum λk satisfying µj ≤ λj ≤ ˜ µj, the sequence of signs νk = ±1.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 10 / 20

Marchenko-Ostrovsky theory

Motivation for the Proposition In order to determine the potential it is enough to know the spectra of the DD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enough to know the functions

t22(λ) - its zeroes form the spectrum of D-N problem; t12(λ) - its zeroes form the spectrum of D-D problem.

The spectrum of the periodic Schr¨

• dinger operator (periodic and

antiperiodic problems) allows one to determine the quasimomentum θ(λ) so that we have u+(λ) = cos θ(λ). The numbers λk give the spectrum the D-D problem, or the function t12(λ). For λ = λk we have: t11 + t22 = 2 cos θ(λk), t11t22 = 1 ⇒ u−(λk) = νk

• u2

+ − 1, νk = ±1.

So in order to determine the D-N spectrum (the function t22) one needs to know the sequence of signs νk.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 11 / 20

Marchenko-Ostrovsky theory

Motivation for the Proposition In order to determine the potential it is enough to know the spectra of the DD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enough to know the functions

t22(λ) - its zeroes form the spectrum of D-N problem; t12(λ) - its zeroes form the spectrum of D-D problem.

The spectrum of the periodic Schr¨

• dinger operator (periodic and

antiperiodic problems) allows one to determine the quasimomentum θ(λ) so that we have u+(λ) = cos θ(λ). The numbers λk give the spectrum the D-D problem, or the function t12(λ). For λ = λk we have: t11 + t22 = 2 cos θ(λk), t11t22 = 1 ⇒ u−(λk) = νk

• u2

+ − 1, νk = ±1.

So in order to determine the D-N spectrum (the function t22) one needs to know the sequence of signs νk.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 11 / 20

Marchenko-Ostrovsky theory

Motivation for the Proposition In order to determine the potential it is enough to know the spectra of the DD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enough to know the functions

t22(λ) - its zeroes form the spectrum of D-N problem; t12(λ) - its zeroes form the spectrum of D-D problem.

The spectrum of the periodic Schr¨

• dinger operator (periodic and

antiperiodic problems) allows one to determine the quasimomentum θ(λ) so that we have u+(λ) = cos θ(λ). The numbers λk give the spectrum the D-D problem, or the function t12(λ). For λ = λk we have: t11 + t22 = 2 cos θ(λk), t11t22 = 1 ⇒ u−(λk) = νk

• u2

+ − 1, νk = ±1.

So in order to determine the D-N spectrum (the function t22) one needs to know the sequence of signs νk.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 11 / 20

Marchenko-Ostrovsky theory

Motivation for the Proposition In order to determine the potential it is enough to know the spectra of the DD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enough to know the functions

t22(λ) - its zeroes form the spectrum of D-N problem; t12(λ) - its zeroes form the spectrum of D-D problem.

The spectrum of the periodic Schr¨

• dinger operator (periodic and

antiperiodic problems) allows one to determine the quasimomentum θ(λ) so that we have u+(λ) = cos θ(λ). The numbers λk give the spectrum the D-D problem, or the function t12(λ). For λ = λk we have: t11 + t22 = 2 cos θ(λk), t11t22 = 1 ⇒ u−(λk) = νk

• u2

+ − 1, νk = ±1.

So in order to determine the D-N spectrum (the function t22) one needs to know the sequence of signs νk.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 11 / 20

Marchenko-Ostrovsky theory

The potential is uniquely determined by the function u+(λ); the function t12(λ); the function u−(λ).

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 12 / 20

Inverse problems for simple graphs Ring

Inverse problems for simple graphs Ring graph Γ1

Φ1 - the total flux through the ring Φ1 = x2

x1 a(y)dy.

Lq,Φ1 - magnetic Schr¨

• dinger operator.

E is an eigenvalue of Lq,Φ1 if and only if it belongs to the interval of the absolutely continuous spectrum of the periodic operator Lper

q

corresponding to the quasimomentum θ = Φ1. The knowledge of En(Φ1) allows one to recover just the function u+(λ) = Tr T(λ)/2. The potential can be reconstructed only in the very exceptional case of zero or constant potential.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 13 / 20

Inverse problems for simple graphs Lassoo

Lassoo graph Γ2

The knowledge of the TW function MΦj(λ, Γ2) by Boundary-Control method allows one to determine the TW function MΦ1(λ, Γ1) (where Γ1 is the ring graph with one contact point) MΦ1(λ, Γ1) = 2 cos Φ1 − Tr T(λ) t12(λ) . The knowldege of the TW matrix for the magnetic flux Φ1 = 0, π (and for all

• ther values of Φ1) allows one to recover just

the function u+(λ) = Tr T(λ)/2; the function t12(λ). To reconstruct the potential on the ring we need to know in addition the sequence of signs νk or, equivalently, the function u−(λ). Reconstruction of the potential on the ring can be carried out, but it is not

• unique. The potential on the boundary edge is uniquely determined by

MΦ(λ, Γ2).

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 14 / 20

Inverse problems for simple graphs Zweih¨ ander

Zweih¨ ander graph Γ3

The knowledge of the TW function MΦj(λ, Γ2) by the Boundary-Control method allows one to determine the 2 × 2 TW function MΦ1(λ, Γ4), where Γ4 is the ring graph with two contact points M(λ, Γ4) = 1 t1

12t2 12

• −(T 1T 2)12

t1

12eiΦ2 + t2 12e−iΦ1

t2

12eiΦ1 + t1 12e−iΦ2

−(T 2T 1)12

• ,

(4) where T 1,2 are the transfer matrices for the two intervals forming the circle. NB! The TW matrix can be reconstructed up to the similarity transformation with diagonal unitary matrix M(λ) = eiΦ3 eiΦ4

• M(λ, Γ4)

e−iΦ3 e−iΦ4

• =

   −(T 1T 2)12

t1

12t2 12

• 1

t2

12 +

1 t1

12 e−iΦ

ei(Φ2+Φ3−Φ4)

• 1

t2

12 +

1 t1

12 eiΦ

e−i(Φ2+Φ3−Φ4) −(T 2T 1)12

t1

12t2 12

   .

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 15 / 20

Inverse problems for simple graphs Zweih¨ ander

Zweih¨ ander graph Γ3

The knowledge of the TW function MΦj(λ, Γ2) by the Boundary-Control method allows one to determine the 2 × 2 TW function MΦ1(λ, Γ4), where Γ4 is the ring graph with two contact points M(λ, Γ4) = 1 t1

12t2 12

• −(T 1T 2)12

t1

12eiΦ2 + t2 12e−iΦ1

t2

12eiΦ1 + t1 12e−iΦ2

−(T 2T 1)12

• ,

(4) where T 1,2 are the transfer matrices for the two intervals forming the circle. NB! The TW matrix can be reconstructed up to the similarity transformation with diagonal unitary matrix M(λ) = eiΦ3 eiΦ4

• M(λ, Γ4)

e−iΦ3 e−iΦ4

• =

   −(T 1T 2)12

t1

12t2 12

• 1

t2

12 +

1 t1

12 e−iΦ

ei(Φ2+Φ3−Φ4)

• 1

t2

12 +

1 t1

12 eiΦ

e−i(Φ2+Φ3−Φ4) −(T 2T 1)12

t1

12t2 12

   .

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 15 / 20

Inverse problems for simple graphs Zweih¨ ander

No-resonance condition

No-resonance condition 1. We say that the no-resonance condition is satisfied if and and only if the D-D spectra of the SL operators on the intervals [x1, x2] and [x3, x4]do not intersect. Necessary and sufficient conditions: There exists an eigenfunction supported by the kernel ⇒ no-resonance condition is violated at this value of the energy. No-resonance condition is violated ⇒

either there exists an eigenfunction supported by the kernel,

• r the scattering matrix is diagonal (at this energy).

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 16 / 20

Inverse problems for simple graphs Zweih¨ ander

No-resonance condition

No-resonance condition 1. We say that the no-resonance condition is satisfied if and and only if the D-D spectra of the SL operators on the intervals [x1, x2] and [x3, x4]do not intersect. Necessary and sufficient conditions: There exists an eigenfunction supported by the kernel ⇒ no-resonance condition is violated at this value of the energy. No-resonance condition is violated ⇒

either there exists an eigenfunction supported by the kernel,

• r the scattering matrix is diagonal (at this energy).

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 16 / 20

Inverse problems for simple graphs Zweih¨ ander

Theorem 1. Let the no-resonance condition be satisfied. Then the potential on Γ3 is uniquely determined by the TW-function M(λ, Γ3) known for Φ = 0, π, where Φ is the total flux of the magnetic field through the ring Φ =

• [x1,x2]∪[x3,x4] a(y)dy.

Idea of the proof

• |(M0(λ))12|

=

• 1

t1

12 +

1 t2

12

• ,

1 4

• |(M0(λ))12|2 − |(Mπ(λ))12|2

=

1 t1

12

1 t2

12 .

⇒ the analytic functions t1

12 and t2 12 are determined.

The entry 11 gives us the function

• T 1(λ)T 2(λ)
• 12 = t1

11(λ)t2 12(λ) + t1 12(λ)t2 22(λ) = −t1 12(λ)t2 12(λ)(M0(λ))11.

Consider the points λ1

j - the zeroes of t1 12

t1

11(λ1 j ) = (T 1(λ1 j )T 2(λ1 j ))12/t2 12(λ1 j )

⇒ the entire function of exponential type t1

11 is uniquely determined ⇒ the

function t1

22 is determined.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 17 / 20

Inverse problems for simple graphs Zweih¨ ander

Theorem 1. Let the no-resonance condition be satisfied. Then the potential on Γ3 is uniquely determined by the TW-function M(λ, Γ3) known for Φ = 0, π, where Φ is the total flux of the magnetic field through the ring Φ =

• [x1,x2]∪[x3,x4] a(y)dy.

Idea of the proof

• |(M0(λ))12|

=

• 1

t1

12 +

1 t2

12

• ,

1 4

• |(M0(λ))12|2 − |(Mπ(λ))12|2

=

1 t1

12

1 t2

12 .

⇒ the analytic functions t1

12 and t2 12 are determined.

The entry 11 gives us the function

• T 1(λ)T 2(λ)
• 12 = t1

11(λ)t2 12(λ) + t1 12(λ)t2 22(λ) = −t1 12(λ)t2 12(λ)(M0(λ))11.

Consider the points λ1

j - the zeroes of t1 12

t1

11(λ1 j ) = (T 1(λ1 j )T 2(λ1 j ))12/t2 12(λ1 j )

⇒ the entire function of exponential type t1

11 is uniquely determined ⇒ the

function t1

22 is determined.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 17 / 20

Inverse problems for simple graphs Zweih¨ ander

Theorem 1. Let the no-resonance condition be satisfied. Then the potential on Γ3 is uniquely determined by the TW-function M(λ, Γ3) known for Φ = 0, π, where Φ is the total flux of the magnetic field through the ring Φ =

• [x1,x2]∪[x3,x4] a(y)dy.

Idea of the proof

• |(M0(λ))12|

=

• 1

t1

12 +

1 t2

12

• ,

1 4

• |(M0(λ))12|2 − |(Mπ(λ))12|2

=

1 t1

12

1 t2

12 .

⇒ the analytic functions t1

12 and t2 12 are determined.

The entry 11 gives us the function

• T 1(λ)T 2(λ)
• 12 = t1

11(λ)t2 12(λ) + t1 12(λ)t2 22(λ) = −t1 12(λ)t2 12(λ)(M0(λ))11.

Consider the points λ1

j - the zeroes of t1 12

t1

11(λ1 j ) = (T 1(λ1 j )T 2(λ1 j ))12/t2 12(λ1 j )

⇒ the entire function of exponential type t1

11 is uniquely determined ⇒ the

function t1

22 is determined.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 17 / 20

Inverse problems for simple graphs Zweih¨ ander

Theorem 1. Let the no-resonance condition be satisfied. Then the potential on Γ3 is uniquely determined by the TW-function M(λ, Γ3) known for Φ = 0, π, where Φ is the total flux of the magnetic field through the ring Φ =

• [x1,x2]∪[x3,x4] a(y)dy.

Idea of the proof

• |(M0(λ))12|

=

• 1

t1

12 +

1 t2

12

• ,

1 4

• |(M0(λ))12|2 − |(Mπ(λ))12|2

=

1 t1

12

1 t2

12 .

⇒ the analytic functions t1

12 and t2 12 are determined.

The entry 11 gives us the function

• T 1(λ)T 2(λ)
• 12 = t1

11(λ)t2 12(λ) + t1 12(λ)t2 22(λ) = −t1 12(λ)t2 12(λ)(M0(λ))11.

Consider the points λ1

j - the zeroes of t1 12

t1

11(λ1 j ) = (T 1(λ1 j )T 2(λ1 j ))12/t2 12(λ1 j )

⇒ the entire function of exponential type t1

11 is uniquely determined ⇒ the

function t1

22 is determined.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 17 / 20

General result

General result

Theorem 2. Assume that: Γ is a metric graph which is:

formed by a finite number of compact intervals, has no loops, has Euler characteristic zero, i.e has one cycle;

Lq,a is the magnetic Schr¨

• dinger operator in L2(Γ), with

q ∈ L2(Γ) real, a ∈ C(Γ) real, standard boundary conditions at the vertices;

Φ is the total flux through the cycle; MΦ(λ) is the TW matrix function. Then the TW matrix function MΦ(λ) known for Φ = 0, π determines the graph Γ and the potential q, provided that the no-resonance condition is satisfied.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 18 / 20

General result

No-resonance condition Let the kernel ker Γ be a cycle divided by contact points γj, j = 1, 2, ..., M into M intervals [x2j−1, x2j]. Denote by Lq,a|j,k

kerΓ, j = k the self-adjoint operator determined by the differential

expression Lq,a on the domain of functions from ⊕ M

j=1 W 2 2 ([x2j−1x2j])

satisfying Dirichlet boundary conditions at the contact vertices γj and γk and the standard boundary conditions at all other contact points. No-resonance condition 2. We say that the no-resonance condition is satisfied if and and only if the spectrum of at least one of the self-adjoint

• perator Lq,a|j,k

kerΓ is simple, i.e. no multiple eigenvalue occurs.

Reconstruction of the potential on the branches

• S. Avdonin, P. K., Inverse problems for quantum trees, Inverse Probl.

Imaging, 2 (2008), no. 1, 1–21. Then Theorem 1 implies Theorem 2.

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 19 / 20

General result

Diolch yn Fawr!

Kurasov (Lund) Inverse Problem for Graphs with Cycles Gregynog 2008 20 / 20