Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
Localization and quantum blockade on graphsand inverse problems for Aharonov-Bohm rings
Pavel Kurasov
Lund University, SWEDEN
May 23, 2009
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 1 / 21
Introduction
1 IntroductionQuantum graphsSpectral propertiesInverse problemsProperties of eigenfucntionsThe main idea
2 Marchenko-Ostrovsky theory
3 Inverse problems for simple graphsRingLassoZweihander
4 General result
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 2 / 21
Introduction Quantum graphs
Quantum graph as a triplet
1 Metric graph Γ - union of intervals ∆j = [x2j−1, x2j ] connected together atthe vertices Vm
⇒ the Hilbert space L2(Γ);
2 Differential expression (formally symmetric) on the edges
Lq,a =
(−1
i
d
dx+ a(x)
)2
+ q(x),
where a, q - real magnetic and electric potentials⇒ the linear operator Lq,a;
3 Matching (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 boundaryconditions, that is:
the function is continuous,
the sum of ”normal” derivatives is zeroKurasov (Lund) Localization and quantum blockade on graphs Warszawa 3 / 21
Introduction Quantum graphs
Quantum graph as a triplet
1 Metric graph Γ - union of intervals ∆j = [x2j−1, x2j ] connected together atthe vertices Vm
⇒ the Hilbert space L2(Γ);
2 Differential expression (formally symmetric) on the edges
Lq,a =
(−1
i
d
dx+ a(x)
)2
+ q(x),
where a, q - real magnetic and electric potentials⇒ the linear operator Lq,a;
3 Matching (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 boundaryconditions, that is:
the function is continuous,
the sum of ”normal” derivatives is zeroKurasov (Lund) Localization and quantum blockade on graphs Warszawa 3 / 21
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
dx2ψ(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).
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 4 / 21
Introduction Inverse problems
Inverse problems:concise historical overview
Solution of the inverse problem for quantum graphs means reconstructionof
the metric graph;
the differential expressions on the edges;
the coupling conditions at the vertices.
Spectral data used
Spectrum of the Schrodinger operator (discrete set of real eigenvaluestending to +∞);
Titchmarsh-Weyl matrix function M(λ) (Dirichlet-to-Neumann map)
M(λ)~u = ∂~u,
where u(λ) is a solution of the differential equation
Lq,au = λu, =λ > 0;
scattering matrix S(λ) associated with the extended graph.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 5 / 21
Introduction Inverse problems
Inverse problems:concise historical overview
Solution of the inverse problem for quantum graphs means reconstructionof
the metric graph;
the differential expressions on the edges;
the coupling conditions at the vertices.
Spectral data used
Spectrum of the Schrodinger operator (discrete set of real eigenvaluestending to +∞);
Titchmarsh-Weyl matrix function M(λ) (Dirichlet-to-Neumann map)
M(λ)~u = ∂~u,
where u(λ) is a solution of the differential equation
Lq,au = λu, =λ > 0;
scattering matrix S(λ) associated with the extended graph.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 5 / 21
Introduction Inverse problems
Obtained results NB! for zero magnetic potential!
Reconstruction of the graph:
with rationally independent lengths (from spectrum):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 (from T-W function):V. Yurko ’06, S. Avdonin and P. K. ’08;calculation of the Euler characteristic (from spectrum):P. K. ’08, ’08;
Reconstruction of the potential on graphs (from T-W function):
star graph:N.I. Gerasimenko and B.S. Pavlov, ’88;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, ’01; V. Yurko, ’09.
Reconstruction of the boundary conditions:
for star graphs:V. Kostrykin and R. Schrader ’00, ’06; M. Harmer ’03.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 6 / 21
Introduction Properties of eigenfucntions
Unusual spectral propertiesof graphs with cycles
Difficulties to solve inverse problems for graphs are related with the properties ofeigenfunctions:
The eigenfunctions may vanish on sets of positive Lebesgue measure ⇒localized eigenfunctions
The Titchmarsh-Weyl matrix function (Dirichlet-to-Neumann map) may bediagonal at certain energies.
These properties are best understood by considering graphs with semiinfiniteedges ⇒ scattering theory.
The operator may have localized eigenfunctions (with compact support)⇒ imbedded eigenvalues are possible.
The transition coefficient may vanish at certain energies ⇒ quantumblockade.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 7 / 21
Introduction The main idea
The main idea
Conclusions concerning recovering the potential
Knowldege of the spectrum alone is not enough to reconstruct thepotential.
Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) isan efficient tool to solve the inverse problem for graphs.
Potential on the branches can be reconstructed from the TW functionusing Boundary Control method.
Potential on the kernel of the graph in general cannot be determined by theTW function.
Our programme
Study the possibility to reconstruct the graph Γ and potential q on itfrom the TW function known for different values of the magnetic field.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 8 / 21
Introduction The main idea
The main idea
Conclusions concerning recovering the potential
Knowldege of the spectrum alone is not enough to reconstruct thepotential.
Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) isan efficient tool to solve the inverse problem for graphs.
Potential on the branches can be reconstructed from the TW functionusing Boundary Control method.
Potential on the kernel of the graph in general cannot be determined by theTW function.
Our programme
Study the possibility to reconstruct the graph Γ and potential q on itfrom the TW function known for different values of the magnetic field.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 8 / 21
Introduction The main idea
The main idea
Conclusions concerning recovering the potential
Knowldege of the spectrum alone is not enough to reconstruct thepotential.
Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) isan efficient tool to solve the inverse problem for graphs.
Potential on the branches can be reconstructed from the TW functionusing Boundary Control method.
Potential on the kernel of the graph in general cannot be determined by theTW function.
Our programme
Study the possibility to reconstruct the graph Γ and potential q on itfrom the TW function known for different values of the magnetic field.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 8 / 21
Introduction The main idea
The main idea
Conclusions concerning recovering the potential
Knowldege of the spectrum alone is not enough to reconstruct thepotential.
Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) isan efficient tool to solve the inverse problem for graphs.
Potential on the branches can be reconstructed from the TW functionusing Boundary Control method.
Potential on the kernel of the graph in general cannot be determined by theTW function.
Our programme
Study the possibility to reconstruct the graph Γ and potential q on itfrom the TW function known for different values of the magnetic field.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 8 / 21
Introduction The main idea
The main idea
Conclusions concerning recovering the potential
Knowldege of the spectrum alone is not enough to reconstruct thepotential.
Titchmarsh-Weyl function (equivalently the Dirichlet-to-Neumann map) isan efficient tool to solve the inverse problem for graphs.
Potential on the branches can be reconstructed from the TW functionusing Boundary Control method.
Potential on the kernel of the graph in general cannot be determined by theTW function.
Our programme
Study the possibility to reconstruct the graph Γ and potential q on itfrom the TW function known for different values of the magnetic field.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 8 / 21
Marchenko-Ostrovsky theory
Marchenko-Ostrovsky theory
Provides necessary and sufficient conditions for a sequence of intervals to be thespectrum of one-dimensional periodic Schrodinger operator Lper
q .Transfer matrix T (a, b; k)
− d2
dx2ψ(x) + q(x)ψ(x) = λψ(x)
⇒ T (a, b; k) =
(t11(k) t12(k)t21(k) t22(k)
):
(ψ(a)ψ′(a)
)7→(
ψ(b)ψ′(b)
), k2 = λ
Introduce the functions:
u±(k) = (t11(k)±t22(k))/2
The end points of the spectral intervals µj , µj are solutions to the equation
u+(k) = ±1⇒ k2 = µj or µj .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 9 / 21
Marchenko-Ostrovsky theory
Proposition 1. 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 n2 [0, π], it is necessary and sufficient that there exist a sequence of real
numbers hj (j = 0,±1,±2, . . . ) satisfying the conditions
∞∑j=1
(jn+1hj)2 <∞, h0 = 0, hj = h−j ≥ 0(j = 1, 2, . . . ), (2)
such that√µj = z(πj − 0),
√µj = z(πj + 0) (j = 1, 2, . . . ),
where the function z(θ) effects a conformal mapping of the region
{θ : Im θ > 0} \ ∪+∞j=−∞ {θ : Re θ = jπ, 0 ≤ Im θ ≤ hj} (3)
into the upper half-plane, normalized by the conditions
θ(0) = 0, limy→∞
(iy)−1θ(iy) = π.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 10 / 21
Marchenko-Ostrovsky theory
In fact the following statement has been proven for example in
V.A. Marchenko, I.V. Ostrovsky, A characterization of the spectrum of theHill operator (Russian) Mat. Sb. (N.S.), 97 (139) (1975), no. 4(8),540–606.
See also: H.P. McKean, P. van Moerbeke, E. Trubowitz, ...
Proposition 2. Assume that all conditions of Proposition 1 are satisfied. Thefollowing 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 λj satisfying µj ≤ λj ≤ µj ,
the sequence of signs νk = ±1.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 11 / 21
Marchenko-Ostrovsky theory
Motivation for the Proposition
In order to determine the potential it is enough to know the spectra of theDD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enoughto know the functions
t22 - its zeroes form the spectrum of D-N problem;t12 - its zeroes form the spectrum of D-D problem.
The spectrum of the periodic Schrodinger operator (periodic andantiperiodic problems) allows one to determine the quasimomentum θ(k)so that we have
u+(k) = cos θ(k), k2 = λ.
The numbers λj give the spectrum of the D-D problem, or the function t12.
For k2j = λj we have:
t11 + t22 = 2 cos θ(kj), t11t22 = 1⇒ u−(kj) = νj
√u2
+ − 1, νj = ±1.
So in order to determine the D-N spectrum (the function t22) one needs toknow the sequence of signs νj .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 12 / 21
Marchenko-Ostrovsky theory
Motivation for the Proposition
In order to determine the potential it is enough to know the spectra of theDD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enoughto know the functions
t22 - its zeroes form the spectrum of D-N problem;t12 - its zeroes form the spectrum of D-D problem.
The spectrum of the periodic Schrodinger operator (periodic andantiperiodic problems) allows one to determine the quasimomentum θ(k)so that we have
u+(k) = cos θ(k), k2 = λ.
The numbers λj give the spectrum of the D-D problem, or the function t12.
For k2j = λj we have:
t11 + t22 = 2 cos θ(kj), t11t22 = 1⇒ u−(kj) = νj
√u2
+ − 1, νj = ±1.
So in order to determine the D-N spectrum (the function t22) one needs toknow the sequence of signs νj .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 12 / 21
Marchenko-Ostrovsky theory
Motivation for the Proposition
In order to determine the potential it is enough to know the spectra of theDD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enoughto know the functions
t22 - its zeroes form the spectrum of D-N problem;t12 - its zeroes form the spectrum of D-D problem.
The spectrum of the periodic Schrodinger operator (periodic andantiperiodic problems) allows one to determine the quasimomentum θ(k)so that we have
u+(k) = cos θ(k), k2 = λ.
The numbers λj give the spectrum of the D-D problem, or the function t12.
For k2j = λj we have:
t11 + t22 = 2 cos θ(kj), t11t22 = 1⇒ u−(kj) = νj
√u2
+ − 1, νj = ±1.
So in order to determine the D-N spectrum (the function t22) one needs toknow the sequence of signs νj .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 12 / 21
Marchenko-Ostrovsky theory
Motivation for the Proposition
In order to determine the potential it is enough to know the spectra of theDD and DN problems (Borg-Levitan-Marchenko). Equivalently it is enoughto know the functions
t22 - its zeroes form the spectrum of D-N problem;t12 - its zeroes form the spectrum of D-D problem.
The spectrum of the periodic Schrodinger operator (periodic andantiperiodic problems) allows one to determine the quasimomentum θ(k)so that we have
u+(k) = cos θ(k), k2 = λ.
The numbers λj give the spectrum of the D-D problem, or the function t12.
For k2j = λj we have:
t11 + t22 = 2 cos θ(kj), t11t22 = 1⇒ u−(kj) = νj
√u2
+ − 1, νj = ±1.
So in order to determine the D-N spectrum (the function t22) one needs toknow the sequence of signs νj .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 12 / 21
Marchenko-Ostrovsky theory
The potential is uniquely determined by
the function u+(λ) = t11(λ) + t22(λ);
the function t12(λ);
the function u−(λ) = t11(λ)− t22(λ).
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 13 / 21
Inverse problems for simple graphs Ring
Inverse problems for simple graphs
Ring graph Γ1
Φ1 - the total flux through the ring Φ1 =∫ x2
x1a(y)dy .
Lq,Φ1 - magnetic Schrodinger operator.
E is an eigenvalue of Lq,Φ1 if and only if it belongs to the interval of theabsolutely continuous spectrum of the periodic operator Lper
q corresponding tothe quasimomentum θ = Φ1.
The knowledge of En(Φ1) allows one to recover just
the function u+(k) = Tr T (k)/2.
The potential can be reconstructed only in the very exceptional case of zero orconstant potential (analog of the result due to Ambartsumian).
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 14 / 21
Inverse problems for simple graphs Lasso
Lasso graph Γ2
The knowledge of the TW function MΦj (λ, Γ2) by Boundary-Control methodallows one to determine the TW function MΦ1 (λ, Γ1) (where Γ1 is the ringgraph with one contact point)
MΦ1 (λ, Γ1) =2 cos Φ1 − Tr T (k)
t12(k).
The knowldege of the TW matrix for the magnetic flux Φ1 = 0, π (and for allother values of Φ1) allows one to recover just
the function u+(k) = Tr T (k)/2;
the function t12(k).
To reconstruct the potential on the ring we need to know in addition thesequence of signs νk or, equivalently, the function u−(k).Reconstruction of the potential on the ring can be carried out, but it is notunique. The potential on the boundary edge is uniquely determined byMΦ(λ, Γ2).
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 15 / 21
Inverse problems for simple graphs Zweihander
Zweihander graph Γ3
The knowledge of the TW function MΦj (λ, Γ2) by the Boundary-Controlmethod allows one to determine the 2× 2 TW function MΦ1 (λ, Γ4), where Γ4 isthe ring graph with two contact points
M(λ, Γ4) =1
t112t2
12
(−(T 1T 2)12 t1
12e iΦ2 + t212e−iΦ1
t212e iΦ1 + 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 transformationwith diagonal unitary matrix
M(λ) =
(e iΦ3 0
0 e iΦ4
)M(λ, Γ4)
(e−iΦ3 0
0 e−iΦ4
)
=
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 16 / 21
Inverse problems for simple graphs Zweihander
Zweihander graph Γ3
The knowledge of the TW function MΦj (λ, Γ2) by the Boundary-Controlmethod allows one to determine the 2× 2 TW function MΦ1 (λ, Γ4), where Γ4 isthe ring graph with two contact points
M(λ, Γ4) =1
t112t2
12
(−(T 1T 2)12 t1
12e iΦ2 + t212e−iΦ1
t212e iΦ1 + 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 transformationwith diagonal unitary matrix
M(λ) =
(e iΦ3 0
0 e iΦ4
)M(λ, Γ4)
(e−iΦ3 0
0 e−iΦ4
)
=
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 16 / 21
Inverse problems for simple graphs Zweihander
Resonance condition and the main theorem
Definition 1. We say that there is a resonance iff the D-D spectra of the SLoperators on the intervals [x1, x2] and [x3, x4] do intersect, i.e. λ1
j = λ2m for
certain j ,m.
Necessary and sufficient conditions for resonance:
Localized eigenfunction with the energy E0 (supported by the kernel) ⇒resonance at the energy E0.
Resonance at the energy E0 ⇒either localized eigenfunction with the energy E0, orthe scattering matrix is diagonal at E0 ⇒ quantum blockade.
Main TheoremAssume that there is no resonance. Then the potential on Γ3
is uniquely determined by the TW-function M(λ, Γ3) known forΦ = 0, π, where Φ is the total flux of the magnetic field throughthe ring Φ =
∫[x1,x2]∪[x3,x4]
a(y)dy .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 17 / 21
Inverse problems for simple graphs Zweihander
Resonance condition and the main theorem
Definition 2. We say that there is a resonance iff the D-D spectra of the SLoperators on the intervals [x1, x2] and [x3, x4] do intersect, i.e. λ1
j = λ2m for
certain j ,m.
Necessary and sufficient conditions for resonance:
Localized eigenfunction with the energy E0 (supported by the kernel) ⇒resonance at the energy E0.
Resonance at the energy E0 ⇒either localized eigenfunction with the energy E0, orthe scattering matrix is diagonal at E0 ⇒ quantum blockade.
Main TheoremAssume that there is no resonance. Then the potential on Γ3
is uniquely determined by the TW-function M(λ, Γ3) known forΦ = 0, π, where Φ is the total flux of the magnetic field throughthe ring Φ =
∫[x1,x2]∪[x3,x4]
a(y)dy .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 17 / 21
Inverse problems for simple graphs Zweihander
Resonance condition and the main theorem
Definition 3. We say that there is a resonance iff the D-D spectra of the SLoperators on the intervals [x1, x2] and [x3, x4] do intersect, i.e. λ1
j = λ2m for
certain j ,m.
Necessary and sufficient conditions for resonance:
Localized eigenfunction with the energy E0 (supported by the kernel) ⇒resonance at the energy E0.
Resonance at the energy E0 ⇒either localized eigenfunction with the energy E0, orthe scattering matrix is diagonal at E0 ⇒ quantum blockade.
Main TheoremAssume that there is no resonance. Then the potential on Γ3
is uniquely determined by the TW-function M(λ, Γ3) known forΦ = 0, π, where Φ is the total flux of the magnetic field throughthe ring Φ =
∫[x1,x2]∪[x3,x4]
a(y)dy .
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 17 / 21
General result
Idea of the proof
M(λ) =
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
{|(M0(λ))12| =
∣∣∣ 1t1
12+ 1
t212
∣∣∣ ,14
(|(M0(λ))12|2 − |(Mπ(λ))12|2
)= 1
t112
1t2
12.
⇒ the analytic functions t112 and t2
12 are determined.The entry 11 gives us the function(
T 1(k)T 2(k))
12= t1
11(k)t212(k) + t1
12(k)t222(k)= −t1
12(k)t212(k)(M0(k2))11.
Consider the points k1j - the zeroes of t1
12
t111(k1
j ) = (T 1(k1j )T 2(k1
j ))12/t212(k1
j )
⇒ the entire function of exponential type t111 is uniquely determined ⇒ the
function t122 is determined.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 18 / 21
General result
Idea of the proof
M(λ) =
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
{|(M0(λ))12| =
∣∣∣ 1t1
12+ 1
t212
∣∣∣ ,14
(|(M0(λ))12|2 − |(Mπ(λ))12|2
)= 1
t112
1t2
12.
⇒ the analytic functions t112 and t2
12 are determined.The entry 11 gives us the function(
T 1(k)T 2(k))
12= t1
11(k)t212(k) + t1
12(k)t222(k)= −t1
12(k)t212(k)(M0(k2))11.
Consider the points k1j - the zeroes of t1
12
t111(k1
j ) = (T 1(k1j )T 2(k1
j ))12/t212(k1
j )
⇒ the entire function of exponential type t111 is uniquely determined ⇒ the
function t122 is determined.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 18 / 21
General result
Idea of the proof
M(λ) =
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
{|(M0(λ))12| =
∣∣∣ 1t1
12+ 1
t212
∣∣∣ ,14
(|(M0(λ))12|2 − |(Mπ(λ))12|2
)= 1
t112
1t2
12.
⇒ the analytic functions t112 and t2
12 are determined.The entry 11 gives us the function(
T 1(k)T 2(k))
12= t1
11(k)t212(k) + t1
12(k)t222(k)= −t1
12(k)t212(k)(M0(k2))11.
Consider the points k1j - the zeroes of t1
12
t111(k1
j ) = (T 1(k1j )T 2(k1
j ))12/t212(k1
j )
⇒ the entire function of exponential type t111 is uniquely determined ⇒ the
function t122 is determined.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 18 / 21
General result
Idea of the proof
M(λ) =
− (T 1T 2)12
t112t
212
(1t2
12+ 1
t112
e−iΦ)
e i(Φ2+Φ3−Φ4)(1t2
12+ 1
t112
e iΦ)
e−i(Φ2+Φ3−Φ4) − (T 2T 1)12
t112t
212
.
{|(M0(λ))12| =
∣∣∣ 1t1
12+ 1
t212
∣∣∣ ,14
(|(M0(λ))12|2 − |(Mπ(λ))12|2
)= 1
t112
1t2
12.
⇒ the analytic functions t112 and t2
12 are determined.The entry 11 gives us the function(
T 1(k)T 2(k))
12= t1
11(k)t212(k) + t1
12(k)t222(k)= −t1
12(k)t212(k)(M0(k2))11.
Consider the points k1j - the zeroes of t1
12
t111(k1
j ) = (T 1(k1j )T 2(k1
j ))12/t212(k1
j )
⇒ the entire function of exponential type t111 is uniquely determined ⇒ the
function t122 is determined.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 18 / 21
General result
Clarifying example
Assume that the Schrodinger operator on the ring Γ3 satisfy the followingconditions:
the intervals [x1, x2] and [x3, x4] have equal lengths;
the potentials q1,2 extended periodically lead to equal band spectra, i.e.
u1+(k) = u2
+(k) ≡ u+(k);
the D-D eigenvalues for q1,2 are also equal, i.e.
t112(k) = t2
12(k) ≡ t12(k).
Then the T-W function is given by
M =1
t12
(−(u1
− − u2−)/2− u+ (e iΦ + 1)e−iφ1
(e−iΦ + 1)e−iφ2 (u1− − u2
−)/2− u+
)It is clear that only the function
u1− − u2
−
can be calculated!Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 19 / 21
General result
Consider again the points kj - zeroes of t12(k).
The values u1,2− (kj) are determined by u+ up to signs ν1,2
k .
The function u1− − u2
− allows one to reconstruct u1,2− (kj) only if
u1−(kj) 6= u2
−(kj).
Consider two potentials q1,2 satisfying our assumptions. These potentialsare uniquely determined by the sequences ν1
k and ν2k .
If ν1k0
= ν2k0
, then let us exchange these signs to opposite ones. The
corresponding potentials q1,2 lead to precisely the same T-W matrix.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 20 / 21
General result
Consider again the points kj - zeroes of t12(k).
The values u1,2− (kj) are determined by u+ up to signs ν1,2
k .
The function u1− − u2
− allows one to reconstruct u1,2− (kj) only if
u1−(kj) 6= u2
−(kj).
Consider two potentials q1,2 satisfying our assumptions. These potentialsare uniquely determined by the sequences ν1
k and ν2k .
If ν1k0
= ν2k0
, then let us exchange these signs to opposite ones. The
corresponding potentials q1,2 lead to precisely the same T-W matrix.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 20 / 21
General result
Consider again the points kj - zeroes of t12(k).
The values u1,2− (kj) are determined by u+ up to signs ν1,2
k .
The function u1− − u2
− allows one to reconstruct u1,2− (kj) only if
u1−(kj) 6= u2
−(kj).
Consider two potentials q1,2 satisfying our assumptions. These potentialsare uniquely determined by the sequences ν1
k and ν2k .
If ν1k0
= ν2k0
, then let us exchange these signs to opposite ones. The
corresponding potentials q1,2 lead to precisely the same T-W matrix.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 20 / 21
General result
Consider again the points kj - zeroes of t12(k).
The values u1,2− (kj) are determined by u+ up to signs ν1,2
k .
The function u1− − u2
− allows one to reconstruct u1,2− (kj) only if
u1−(kj) 6= u2
−(kj).
Consider two potentials q1,2 satisfying our assumptions. These potentialsare uniquely determined by the sequences ν1
k and ν2k .
If ν1k0
= ν2k0
, then let us exchange these signs to opposite ones. The
corresponding potentials q1,2 lead to precisely the same T-W matrix.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 20 / 21
General result
Consider again the points kj - zeroes of t12(k).
The values u1,2− (kj) are determined by u+ up to signs ν1,2
k .
The function u1− − u2
− allows one to reconstruct u1,2− (kj) only if
u1−(kj) 6= u2
−(kj).
Consider two potentials q1,2 satisfying our assumptions. These potentialsare uniquely determined by the sequences ν1
k and ν2k .
If ν1k0
= ν2k0
, then let us exchange these signs to opposite ones. The
corresponding potentials q1,2 lead to precisely the same T-W matrix.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 20 / 21
General result
General result
Theorem 5. 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 Schrodinger 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 graphΓ and the potential q, provided that the no-resonance condition is satisfied.
Kurasov (Lund) Localization and quantum blockade on graphs Warszawa 21 / 21