Localization and quantum blockade on graphs and inverse...

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

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 =

dx+ a(x)

+ 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 =

dx+ a(x)

+ 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

x2n−1

a(y)dy

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

which allows one to eliminate the magnetic field

((−1

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).

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.

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.

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.

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.

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.

The main idea

Our programme

The main idea

Our programme

The main idea

Our programme

The main idea

Our programme

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 .

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.

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.

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

+ − 1, νj = ±1.

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

u+(k) = cos θ(k), k2 = λ.

+ − 1, νj = ±1.

u+(k) = cos θ(k), k2 = λ.

+ − 1, νj = ±1.

u+(k) = cos θ(k), k2 = λ.

+ − 1, νj = ±1.

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).

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).

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).

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

(−(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

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

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

(−(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

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

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 .

j = λ2m for

certain j ,m.

∫[x1,x2]∪[x3,x4]

a(y)dy .

j = λ2m for

certain j ,m.

∫[x1,x2]∪[x3,x4]

a(y)dy .

General result

Idea of the proof

M(λ) =

− (T 1T 2)12

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

{|(M0(λ))12| =

∣∣∣ 1t1

∣∣∣ ,14

(|(M0(λ))12|2 − |(Mπ(λ))12|2

⇒ 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

t111(k1

j ) = (T 1(k1j )T 2(k1

j ))12/t212(k1

⇒ the entire function of exponential type t111 is uniquely determined ⇒ the

function t122 is determined.

General result

Idea of the proof

M(λ) =

− (T 1T 2)12

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

{|(M0(λ))12| =

∣∣∣ 1t1

∣∣∣ ,14

(|(M0(λ))12|2 − |(Mπ(λ))12|2

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.

t111(k1

j ) = (T 1(k1j )T 2(k1

j ))12/t212(k1

General result

Idea of the proof

M(λ) =

− (T 1T 2)12

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

{|(M0(λ))12| =

∣∣∣ 1t1

∣∣∣ ,14

(|(M0(λ))12|2 − |(Mπ(λ))12|2

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.

t111(k1

j ) = (T 1(k1j )T 2(k1

j ))12/t212(k1

General result

Idea of the proof

M(λ) =

− (T 1T 2)12

e−iΦ)

e i(Φ2+Φ3−Φ4)(1t2

e iΦ)

e−i(Φ2+Φ3−Φ4) − (T 2T 1)12

{|(M0(λ))12| =

∣∣∣ 1t1

∣∣∣ ,14

(|(M0(λ))12|2 − |(Mπ(λ))12|2

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.

t111(k1

j ) = (T 1(k1j )T 2(k1

j ))12/t212(k1

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

(−(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

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.

General result

u1−(kj) 6= u2

−(kj).

k and ν2k .

If ν1k0

= ν2k0

General result

u1−(kj) 6= u2

−(kj).

k and ν2k .

If ν1k0

= ν2k0

General result

u1−(kj) 6= u2

−(kj).

k and ν2k .

If ν1k0

= ν2k0

General result

u1−(kj) 6= u2

−(kj).

k and ν2k .

If ν1k0

= ν2k0

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.

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