Riemann--Hilbert problem approach for two-dimensional flow inverse scattering

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Riemann–Hilbert problem approach for two-dimensional flow inverse scattering

Alexey Agaltsov, Roman Novikov

To cite this version:

Alexey Agaltsov, Roman Novikov. Riemann–Hilbert problem approach for two-dimensional flow in-

verse scattering. Journal of Mathematical Physics, American Institute of Physics (AIP), 2014, 55 (10),

pp.103502. �hal-00939283�

two-dimensional ow inverse sattering

∗

A.D.Agaltsov 1

,R.G.Novikov 2

Abstrat

We onsider inverse sattering for the time-harmoni wave equa-

tionwithrst-order perturbation intwodimensions. This problem

arisesinpartiularin theaoustitomographyofmovinguid. We

onsider linearized and nonlinearizedreonstrution algorithms for

this problem of inverse sattering. Our nonlinearized reonstru-

tionalgorithm isbasedon thenon-loal RiemannHilbert problem

approah. Comparisonswithpreeding resultsaregiven.

Keywords: aoustitomography,movinguid,waveequationwith

rst-order perturbation, inverse sattering, RiemannHilbert prob-

lem.

1 Introdution

Weonsider theequation

− ∆ψ − 2iA(x) ∇ ψ + V (x)ψ = Eψ, x = (x

1

, x

2

) ∈ R

²

, E > 0,

^(1.1)

where

∆ = ∂

_x²1

+ ∂

_x²2^,

∇ = (∂

x1

, ∂

x2

)

^,

∂

xk

= ∂/∂x

k^,

k = 1

^,

2

^,and

A = (A

1

, A

2

)

and

V

^{are vetor and salar potentials on}

R

², respetively. In addition we assumethat

A

1^,

A

2 ^and

V

^{are suiently regular funtions on}

R

²

with suientdeayatinnity.

(1.2)

Equation (1.1) anbe onsidered asa model equation for the time-harmoni

exp( − iωt)

^{aoustipressure}

ψ

^inatwo-dimensionalmovinguid,seee.g.[RW℄, [RE℄.Inthissetting

E = ω

c

0

2

, A(x) = ω c

0

u(x), V (x) = 1 − n

²

(x) ω c

0

2

,

^(1.3)

where

c

0 ^{isareferenesoundspeed,}

n(x)

^{isasalarindexofrefration,}

u(x)

^is

anormalizeduidveloityvetor.

∗

ThemainpartoftheworkwasfullledduringthestageoftherstauthorintheCentre

deMathematiquesAppliqueesofEolePolytehniqueinOtoberDeember2013

1

119991,LomonosovMosowStateUniversity,FaultyofComputationalMathematisand

Cybernetis,Mosow,Russia;email: agaletsgmail.om

2

CNRS(UMR7641),CentredeMathematiques Appliquees,EolePolytehnique, 91128

Palaiseau, Frane; IEPTRAS, 117997, Mosow, Russia; Mosow Institute of Physis and

Tehnology,Dolgoprudny,Russia;email: novikovmap.polytehnique.fr

equationatxed energy

E

^{withmagnetipotential}

A

^{andeletripotential}

v

^,

where

V (x) = A

²

(x) − i div A(x) + v(x)

= A

²₁

(x) + A

²₂

(x) − i

∂A

1

(x)

∂x

1

+ ∂A

2

(x)

∂x

2

+ v(x),

(1.4)

seee.g. [HN1℄,[ER2℄.

Forequation(1.1)weonsiderthelassialsatteringsolutions

ψ

^+ontinu-

ousandboundedon

R

²withtheirrstderivativesandspeiedbythefollowing asymptotisas

| x | → ∞

^:

ψ

⁺

(x, k) = e

^ikx

+ C( | k | ) e

^i|k||x|

| x |

^1/2

f

k, | k | x

| x |

+ o 1

| x |

^1/2

,

^(1.5)

x ∈ R

²

, k ∈ R

²

, k

²

= E, C( | k | ) = − πi √

2πe

^−iπ/4

| k |

^1/2

,

where apriori unknownfuntion

f = f (k, l)

^,

k

^,

l ∈ R

²,

k

²

= l

²

= E

^{, arising}

in(1.5)isthelassialsatteringamplitudefor(1.1).

Given potentials

A

^,

V

^{, to determine}

ψ

^{+ and}

f

^{one an use the following}

LippmannShwingerintegralequation(1.6)andformula(1.8)(seee.g. [HN1℄):

ψ

⁺

(x, k) = e

^ikx

+ Z

R²

G

⁺

(x − y, k) ×

× − 2iA(y) ∇

^y

ψ

⁺

(y, k) + V (y)ψ

⁺

(y, k) dy,

(1.6)

G

⁺

(x, k) = − (2π)

⁻²

Z

R²

e

^iξx

dξ

ξ

²

− k

²

− i0 = − i

4 H

₀¹

( | x || k | ),

^(1.7)

where

x ∈ R

²,

k ∈ R

²,

k

²

= E

^,

H

₀^{1 istheHankelfuntionoftherstkind;}

f (k, l) = (2π)

⁻²

Z

R²

e

^−ily

− 2iA(y) ∇

y

ψ

⁺

(y, k) + V (y)ψ

⁺

(y, k)

dy,

^(1.8)

where

k ∈ R

²,

l ∈ R

²,

k

²

= l

²

= E

^{. Atually, we onsider (1.6) and its}

dierentiatedversion,where

∇

^{isappliedtobothsidedof(1.6),asasystemof}

linearintegralequationsfor

ψ

^+and

∇ ψ

^+.

Oneanseethatthesatteringamplitude

f

^{forequation(1.1)at xed}

E

^is

denedon

M

E

=

k ∈ R

²

, l ∈ R

²

: k

²

= l

²

= E , E > 0.

^(1.9)

Notethat

f

^on

M

E ^{isinvariantwithrespettothegauge}transformations

A → A + ∇ ϕ,

V → V − i∆ϕ + ( ∇ ϕ)

²

+ 2A ∇ ϕ,

(1.10)

where

ϕ

^{isasuientlyregularfuntion on}

R

²withsuientdeayatinnity, seee.g. [HN2℄,[ER2℄. Inaddition,

ψ

^{+ is}transformedas

ψ

⁺

→ e

^−iϕ

ψ

^{+ (1.11)}

Inthis work weonsider thefollowinginverse satteringproblem forequa-

tion(1.1)underassumptions(1.2):

Problem 1.1. Given satteringamplitude

f

^on

M

E ^{at xed}

E > 0

^{, nd po-}

tentials

A

^and

V

^on

R

² (atleastapproximately).

Problem 1.1 for the ase when

A ≡ 0

^{wasstudied, in partiular, in [N1℄,}

[GM℄,[N3℄,[GN2℄,[N4℄, [BBMRS℄,[BAR℄andin [N2℄,[Buk℄.

Problem1.1 forthegeneralasewasonsidered,inpartiular, onpage457

of[N3℄andin[X℄.

Problem1.1isalsorelatedwithseveralotherinverseproblemsfortheShrodinger

equationinmagnetield(andforthetime-harmoniwaveequationwithrst-

orderperturbation)in dimension

d ≥ 2

^{. Conerning these otherinverseprob-}

lemssee[DKN℄,[Sh℄,[HN2℄,[Nor℄,[RW℄,[RE℄,[BBS℄,[ER1℄,[NSU℄,[A℄,[ER2℄,

[Ni℄,[WY℄,[GT℄,[IY℄andreferenestherein.

Notethatapproximatending

A

^and

V

^{inProblem1.1means,inpartiular,}

ndingmodulotransformations(1.10) . However,forreal-valued

A

^and

V

^there

is no gauge nonuniqueness (1.10) in Problem 1.1! In addition,

A

^and

V

^of

formulas(1.3)(ofmovinguidaoustis)arerealif

n

^isreal.

Inthisworkwearemainlymotivatedbyappliationstotheaoustitomog-

raphyofmovinguiddisribedin [Nor℄,[RW℄,[RE℄, [BBS℄.Notethat intheir

reonstrution results works [Nor℄, [RW℄, [RE℄, [BBS℄ proeed from near-eld

sattering data (e.g. from some near eld information on

ψ

^{+) instead of the}

satteringamplitude

f

^{. Butitisalsoknownthatnear-eldsatteringdataan}

betransformedintovaluesof

f

^{,see e.g. [Ber℄,[BBS℄.}

Resultsofthepresentworkanbedesribedasfollows:

InSetion 2wegiveformulasforsolvingProblem1.1 inthe Bornapproxi-

mation. Toourknowledgethese formulaswerenotyet givenexpliitely in the

literaturefortheasewhen

A 6≡ 0

^{. Theseformulasareprovedin Setion5.}

In Setion 3wegive anonlinearizedreonstrution algorithm for Problem

1.1. For the ase when

A ≡ 0

^{this algorithm is redued to the algorithm of}

[N4℄and wasimplementednumerially in [BAR℄. Forthegeneral asethis al-

gorithmanbealsoregardedassimpliationanddevelopmentofthealgorithm

mentioned on page 457 of [N3℄ and based on the RiemannHilbertManakov

problemapproahof[GN1℄,[N1℄. Aderivationofthereonstrutionalgorithm

ofSetion3isgivenin Setion6.

InSetion4weshowthatin theBornapproximationthealgorithm ofSe-

tion3isreduedtoformulasofSetion2. RelatedproofsaregiveninSetion7.

Inasimilarwaywithresultsof[NS℄thereonstrutionalgorithmofSetion3

anbegeneralizedto themulti-hannelase. Thisgeneralizationwill begiven

inasubsequentwork.

InthepresentworkwearefousedonapproximatereonstrutionsforProb-

lem 1.1, admitting stable numerial implementation. Issues related with the-

orems of uniqueness and examples of nonuniqueness for Problem 1.1 will be

onsideredinasubsequentwork.

If

A = (A

1

, A

2

)

^and

V

^{satisfy(1.2)andare suientlysmall, thenproeeding}

from(1.6),(1.8)wehavethefollowingBornapproximationformulasfordiret

sattering:

ψ

⁺

(x, k) ≈ e

^ikx

, ∇ ψ

⁺

(x, k) ≈ e

^ikx

ik, f (k, l) ≈ f

^lin

(k, l),

(2.1)

f

^lin

(k, l) == (2π)

^{def −2}

Z

R²

e

^i(k−l)x

2kA(x) + V (x)

dx,

^(2.2)

where

x

^,

k

^,

l ∈ R

²,

k

²

= l

²

= E

^{. Inpartiular, formulas(2.1)anbespeied}

as(2.14).

Notethat

f

^linon

M

E ^{isinvariantwithrespettothegauge}transformations

A → A + ∇ ϕ, V → V − i∆ϕ,

^(2.3)

where

ϕ

^{isasuientlyregularfuntion on}

R

²withsuientdeayatinnity.

This invariane follows from (2.2), (2.3), integrating by parts and using that

k

²

− l

²

= 0

^{. Weonsider(2.3)asa}linearizationof (1.10)forsmall

A

^,

V

^and

ϕ

^.

Inthissetionweonsiderthefollowinglinearizedinversesatteringproblem

forequation(1.1)under assumptions(1.2) :

Problem2.1. Givenlinearizedsatteringamplitude

f

^linon

M

E^atxed

E > 0

^,

ndpotentials

A

^and

V

^on

R

² (atleastapproximately).

Notethatapproximatending

A

^and

V

^{inProblem2.1means,inpartiular,}

ndingmodulotransformations(2.3). However,inasimilarwaywithProblem

1.1, thereis nogaugenonuniqueness (2.3)in Problem 2.1 forthease ofreal-

valued

A

^and

V

^.

Problem2.1isalinearizationofProblem1.1.

TostudyProblem 2.1 itis onvenientto introdue

ϕ

^div,

A

^div,0,

V

^{div,0 and}

ϕ

^±,

A

^±,0,

V

^±,0suhthat:

∆ϕ

^div

(x) = − div A(x), ϕ

^div

(x) → 0

^as

| x | → ∞ , A

^div,0

(x) = A(x) + ∇ ϕ

^div

(x), V

^div,0

(x) = V (x) − i∆ϕ

^div

(x),

(2.4)

where

x ∈ R

²;

∂

z

ϕ

⁻

(x) = − 1

2 (A

1

(x) − iA

2

(x)), ϕ

⁻

(x) → 0

^as

| x | → ∞ , A

^−,0

(x) = A(x) + ∇ ϕ

⁻

(x), V

^−,0

(x) = V (x) − i∆ϕ

⁻

(x),

(2.5)

∂

¯z

ϕ

⁺

(x) = − 1

2 (A

1

(x) + iA

2

(x)), ϕ

⁺

(x) → 0

^as

| x | → ∞ , A

^+,0

(x) = A(x) + ∇ ϕ

⁺

(x), V

^+,0

(x) = V (x) − i∆ϕ

⁺

(x),

(2.6)

∂

z

= 1

2 (∂

x1

− i∂

x2

), ∂

z¯

= 1

2 (∂

x1

+ i∂

x2

), x = (x

1

, x

2

) ∈ R

²

.

^(2.7)

Oneanseethat

div A

^div,0

(x) = 0, A

⁻₁^,0

(x) − iA

⁻₂^,0

(x) = 0, A

^+,0₁

(x) + iA

^+,0₂

(x) = 0,

where

x ∈ R

²,

A

^±,0

= (A

^±₁^,0

, A

^±₂^,0

)

^.

Itisalsoonvenienttotransformformula(2.2)to theform

f

^lin

(k, l) − f

^lin

( − l, − k) = 2(k + l) A(k b − l),

f

^lin

(k, l) + f

^lin

( − l, − k) = 2(k − l) A(k b − l) + 2 V b (k − l),

(2.8)

A(p) = (2π) b

⁻²

Z

R²

e

^ipx

A(x) dx, V b (p) = (2π)

⁻²

Z

R²

e

^ipx

V (x) dx,

^(2.9)

where

(k, l) ∈ M

E^,

p ∈ R

². Notethat

(k, l) ∈ M

E

= ⇒ k − l ∈ B

₂^√_E

,

p ∈ B

₂^√_E

= ⇒ p = k − l

^forsome

(k, l) ∈ M

E

,

(2.10)

where

B

r

= { p ∈ R

²

: | p | ≤ r } , r > 0.

^(2.11)

Wedene

C

^N,σ

( R

²

) =

u ∈ C

^N

( R

²

) : k u k

N,σ

< + ∞ , k u k

N,σ

= max

|n|6N

sup

x∈R²

(1 + | x |

²

)

^σ/2

| ∂

ⁿ

u(x) | , N ∈ N ∪ { 0 } , σ > 0,

^(2.12)

where

C

^N

( R

²

)

^isthespaeof

N

^{-timesontinously}dierentiableomplex-valued funtionson

R

²,

∂

ⁿ

= ∂

_xⁿ1¹

∂

_xⁿ2²

, n = (n

1

, n

2

) ∈ N ∪ { 0 }

2

, | n | = n

1

+ n

2

.

^(2.13)

Notethatif

A

1^,

A

2^,

V ∈ C

^0,σ

( R

²

)

^forsome

σ > 2

^and

k A

j

k

^0,σ

≤ q

^,

j = 1

^,

2

^,

k V k

^0,σ

≤ q

^{, then}

ψ

⁺

(x, k) = e

^ikx

+ O(q), ∇ ψ

⁺

(x, k) = e

^ikx

ik + O(q), f (k, l) = f

^lin

(k, l) + O(q

²

)

^as

q → +0,

(2.14)

uniformlywithrepsetto

x

^,

k

^,

l ∈ R

²,

k

²

= l

²

= E

^,atxed

E > 0

^.

Theorem 2.1. Suppose that

A

1^,

A

2^,

V

^are real-valued and

A

1^,

A

2^,

V ∈

C

^N,σ

( R

²

)

^{for some}

N ≥ 3

^,

σ > 2

^{. Then the following formulas for solving}

A(k b − l) = f

^lin

(k, l) − f

^lin

(l, k) 2

k − l

| k − l |

²

+ f

^lin

(k, l) − f

^lin

( − l, − k) 2

k + l

| k + l |

²

, V b (k − l) = f

^lin

(l, k) + f

^lin

( − l, − k)

2 ,

^(2.15)

where

A b

^,

V b

^{are denedby (2.9)and}

(k, l) ∈ M

E^;

A(x) = A

^appr

(x, E) + A

^err

(x, E), x ∈ R

²

, E > 0, A

^appr

(x, E) =

Z

|p|≤2√ E

e

^−ipx

A(p) b dp, A

^err

(x, E) = Z

|p|≥2√ E

e

^−ipx

A(p) b dp,

^(2.16)

V (x) = V

^appr

(x, E) + V

^err

(x, E), x ∈ R

²

, E > 0,

V

^appr

(x, E) = Z

|p|≤2√ E

e

^−ipx

V b (p) dp, V

^err

(x, E) = Z

|p|≥2√ E

e

^−ipx

V b (p) dp,

^(2.17)

where

A(p) b

^and

V b (p)

^for

| p | ≤ 2 √

E

^{aregiven intermsof}

f

^{lin on}

M

E ^aording

to (2.10), (2.15)and

| A

err,j

(x, E) | ≤ c

1

(N, σ) k A

j

k

^N,σ

E

^−N2−2

,

^(2.18)

| V

^err

(x, E) | ≤ c

1

(N, σ) k V k

^N,σ

E

^−N2−2

,

^(2.19)

where

x ∈ R

²,

j = 1

^,

2

^,

A

^err

= (A

^err,1

, A

^err,2

)

^,

E ≥

¹₄ ^and

c

1

(N, σ) = 4

(N − 2)(σ − 2) .

^(2.20)

Theorem2.2. Supposethat

A

1^,

A

2^,

V ∈ C

^N,σ

( R

²

)

^forsome

N ≥ 4

^and

σ > 2

^.

Let

A

^div,0,

V

^{div,0 bedenedaordingto (2.4). Thenthe followingformulas for}

solvingProblem 2.1arevalid:

b

A

^div,0

(k − l) = f

^lin

(k, l) − f

^lin

( − l, − k) 2

k + l

| k + l |

²

, V b

^div,0

(k − l) = f

^lin

(k, l) + f

^lin

( − l, − k)

2 ,

(2.21)

where

A b

^div,0,

V b

^{div,0 aretheFouriertransformsof}

A

^div,0,

V

^{div,0 (see (2.9))and}

(k, l) ∈ M

E^;

A

^div,0

(x) = A

^div,0appr

(x, E) + A

^div,0err

(x, E), x ∈ R

²

, E > 0,

^(2.22)

A

^div,0appr

(x, E) =

Z

|p|≤2√ E

e

^−ipx

A b

^div,0

(p) dp, A

^div,0err

(x, E) = Z

|p|≥2√ E

e

^−ipx

A b

^div,0

(p) dp, V

^div,0

(x) = V

appr^div,0

(x, E) + V

err^div,0

(x, E), x ∈ R

²

, E > 0,

^(2.23)

V

appr^div,0

(x, E) =

Z

|p|≤2√ E

e

^−ipx

V b

^div,0

(p) dp, V

err^div,0

(x, E) = Z

|p|≥2√ E

e

^−ipx

V b

^div,0

(p) dp,

where

A b

^div,0

(p)

^and

V b

^div,0

(p)

^for

| p | ≤ 2 √

E

^{are given in termsof}

f

^{lin on}

M

E

aording to(2.10) , (2.21)and

| A

^div,0err,j

(x, E) | ≤ (1 + √

2)c

1

(N, σ) k A k

^N,σ

E

^−N−22

,

^(2.24)

| V

err^div,0

(x, E) | ≤ c

1

(N, σ)

k V k

^N,σ

E

^−N2−2

+ √

2 k A k

^N,σ

E

^−N2−3

,

^(2.25)

k A k

^N,σ

= max k A

1

k

^N,σ

, k A

2

k

^N,σ

,

^(2.26)

where

x ∈ R

²,

j = 1

^,

2

^,

E ≥

¹4^,

A

^div,0err

= (A

^div,0err,1

, A

^div,0err,2

)

^and

c

1

(N, σ)

^isdened

by (2.20). Furthermore, if

div A = 0

^then

A

^div,0

= A

^,

V

^div,0

= V

^.

Theorem2.3. Supposethat

A

1^,

A

2^,

V ∈ C

^N,σ

( R

²

)

^forsome

N ≥ 4

^and

σ > 2

^.

Let

A

^±,0,

V

^{±,0 bedenedaording to}(2.5)(2.6). Then thefollowingformulas forsolving Problem 2.1arevalid:

A b

^±₁^,0

(k − l) = 1 2

f (k, l) − f ( − l, − k)

k

1

+ l

1

± i(k

2

+ l

2

) , A b

^±₂^,0

(k − l) = ± i A b

^±₁^,0

(k − l), V b

^±,0

(k − l) = (l

1

± il

2

)f (k, l) + (k

1

± ik

2

)f ( − l, − k)

k

1

+ l

1

± i(k

2

+ l

2

) ,

(2.27)

where

A b

^±,0,

V b

^{±,0 are the Fourier transforms of}

A

^±,0,

V

^{±,0 (see (2.9)) and}

(k, l) ∈ M

E^;

A

^±,0

(x) = A

^±appr^,0

(x, E) + A

^±err^,0

(x, E), x ∈ R

²

, E > 0,

^(2.28)

A

^±appr^,0

(x, E) =

Z

|p|≤2√ E

e

^−ipx

A b

^±,0

(p) dp, A

^±err^,0

(x, E) = Z

|p|≥2√ E

e

^−ipx

A b

^±,0

(p) dp, V

^±,0

(x) = V

appr^±,0

(x, E) + V

err^±,0

(x, E), x ∈ R

²

, E > 0,

^(2.29)

V

appr^±,0

(x, E) =

Z

|p|≤2√ E

e

^−ipx

V b

^±,0

(p) dp, V

err^±,0

(x, E) = Z

|p|≥2√ E

e

^−ipx

V b

^±,0

(p) dp,

where

A b

^±,0

(p)

^and

V b

^±,0

(p)

^for

| p | ≤ 2 √

E

^{are given in terms of}

f

^{lin on}

M

E

aording to(2.10) , (2.27)and

| A

^±err,j^,0

(x, E) | ≤ (1 + √

2)c

1

(N, σ) k A k

N,σ

E

^−N−22

,

^(2.30)

| V

err^±,0

| ≤ c

1

(N, σ)

k V k

^N,σ

E

^−N2−2

+ √

2 k A k

^N,σ

E

^−N−23

,

^(2.31)

where

x ∈ R

²,

j = 1

^,

2

^,

A

^±err^,0

= (A

^±err,1^,0

, A

^±err,2^,0

)

^,

k A k

^{N,σ is dened by (2.23)}

and

c

1

(N, σ)

^{isgiven by (2.20). F}urthermore, if

A

1

± iA

2

= 0

^then

A = A

^±,0,

V = V

^±,0.

Theorems2.12.3areprovedin Setion5.

3.1. Some notations. Tostudy Problem1.1 itis onvenienttointrodue

ϕ

^div,

A

^div,

V

^divand

ϕ

^±,

A

^±,

V

^±,where

ϕ

^divand

ϕ

^{± aredenedaordingto(2.4)}

(2.6)and

A

^div

= A + ∇ ϕ

^div

, V

^div

= V − i∆ϕ

^div

+ ( ∇ ϕ

^div

)

²

+ 2A ∇ ϕ

^div

, A

^±

= A + ∇ ϕ

^±

, V

^±

= V − i∆ϕ

^±

+ ( ∇ ϕ

^±

)

²

+ 2A ∇ ϕ

^±

.

(3.1)

Inthis setion wegiveanonlinearizedalgorithm for approximate nding

A

^±,

V

^{± and}

A

^div,

V

^{div on}

R

² from

f

^on

M

E^{. This algorithm takesinto aount}

multiplesatteringeetsandanberegardedasanonlinearversionofformulas

for

A

^±appr^,0 ,

V

appr^±,0

,

A

^div,0appr ,

V

appr^div,0

of (2.28),(2.29),(2.22),(2.23) .

Itisonvenienttousethefollowingnotations:

z = x

1

+ ix

2

, z ¯ = x

1

− ix

2

,

^(3.2)

λ = E

^−1/2

(k

1

+ ik

2

), λ

^′

= E

^−1/2

(l

1

+ il

2

),

^(3.3)

where

x = (x

1

, x

2

) ∈ R

²,

k = (k

1

, k

2

) ∈ Σ

E^,

l = (l

1

, l

2

) ∈ Σ

E^,

Σ

E

=

m = (m

1

, m

2

) ∈ C

²

: m

²₁

+ m

²₂

= E , E > 0.

^(3.4)

Inthese notations

k

1

= 1

2 E

^1/2

(λ + λ

⁻¹

), k

2

= i

2 E

^1/2

(λ

⁻¹

− λ),

^(3.5)

l

1

= 1

2 E

^1/2

(λ

^′

+ λ

^′−1

), l

2

= i

2 E

^1/2

(λ

^′−1

− λ

^′

),

^(3.6)

exp(ikx) = exp

i

2 E

^1/2

(λ¯ z + λ

⁻¹

z)

,

^(3.7)

where

λ

^,

λ

^′

∈ C \ { 0 }

^,

z ∈ C

²,

k

^,

l ∈ Σ

E^.

Inaddition,usingformulas(1.9),(3.3),(3.4),(3.5),(3.6)oneansee that

Σ

E

∼ = C \ { 0 } , Σ

E

∩ R

²

= S

¹√

E

∼ = T, M

E

∼ = T × T,

(3.8)

where

S

¹_r

=

m ∈ R

²

: | m | = r , r > 0, T =

λ ∈ C : | λ | = 1 .

(3.9)

Inaddition,thefuntions

ψ

^+,

f

^{of(1.5)(1.8)anbewritten as}

ψ

⁺

= ψ

⁺

(z, λ, E), f = f (λ, λ

^′

, E),

^(3.10)

where

λ

^,

λ

^′

∈ T

^,

z ∈ C

,

E > 0

^.

nding

A

^±,

V

^{± and}

A

^div,

V

^{div on}

R

² from

f

^on

M

E ^{hasthefollowingsheme}

f −→ h

_±

−→ µ

⁺

−→ µ

_±

−→ A

^±appr

, V

appr^±

−→ A

^divappr

, V

appr^div

(3.11)

andonsistsofthefollowingsteps:

Step1. Find funtions

h

_±

(λ, λ

^′

, E)

^,

λ

^,

λ

^′

∈ T

^{, from thefollowing linearintegral}

equations:

h

_±

(λ, λ

^′

, E) − πi Z

T

h

_±

(λ, λ

^′′

, E)χ

± i λ

λ

^′′

− λ

^′′

λ

×

× f (λ

^′′

, λ

^′

, E) | dλ

^′′

| = f (λ, λ

^′

, E),

(3.12)

where

χ(s) =

1

^for

s ≥ 0,

0

^for

s < 0

^{. (3.13)}

Step2. Solvethefollowinglinearintegralequationfor

µ

⁺

(z, λ, E)

^,

z ∈ C

,

λ ∈ T

^,

E > 0

^:

µ

⁺

(z, λ, E) + Z

T

B(λ, λ

^′

, z, E)µ

⁺

(z, λ

^′

, E) | dλ

^′

| = 1,

^(3.14)

where

B(λ, λ

^′

, z, E) = 1 2

Z

T

h

₋

(ζ, λ

^′

, z, E)χ

− i ζ

λ

^′

− λ

^′

ζ

dζ ζ − λ(1 − 0) −

− 1 2

Z

T

h

+

(ζ, λ

^′

, z, E)χ

i ζ

λ

^′

− λ

^′

ζ

dζ

ζ − λ(1 + 0) ,

^(3.15)

h

_±

(λ, λ

^′

, z, E) ==

^def

h

_±

(λ, λ

^′

, E) ×

× exp

− i

√ E

2 (λ − λ

^′

)¯ z + (λ

⁻¹

− λ

^′−1

)z ,

(3.16)

and

λ

^,

λ

^′

∈ T

^,

z ∈ C

,

E > 0

^.

Step3. Denefuntions

µ

_±

(z, λ, E)

^,

z ∈ C

,

λ ∈ T

^,

E > 0

^,byformulas

µ

_±

(z, λ, E) = µ

⁺

(z, λ, E) + πi

Z

T

h

_±

(λ, λ

^′

, z, E) ×

× χ

± i λ

λ

^′

− λ

^′

λ

µ

⁺

(z, λ

^′

, E) | dλ

^′

| ,

(3.17)

where funtions

h

_±

(λ, λ

^′

, z, E)

^{are given by (3.16) and}

χ

^{is dened by}

(3.13).

Step4. Funtions

A

^±appr,j

(x, E)

^,

V

appr^±

(x, E)

^,

x ∈ R

²,

j = 1

^,

2

^,

E > 0

^{, aredened}

byformulas

A

⁻appr,1

(x, E) = i

4 a

⁻_z

(z, E ), A

⁻appr,2

(x, E) = 1

4 a

⁻_z

(z, E), a

⁻_z

(z, E) = 4∂

z¯

ln

Z

T

µ

+

(z, ζ, E) | dζ | ,

V

appr⁻

(x, E) =

√ E π

Z

T

∂

z

µ

₋

(z, ζ, E) dζ,

(3.18)

and

A

⁺appr,1

(x, E) = i

4 a

⁺_z¯

(z, E), A

⁺appr,2

(x, E) = − 1

4 a

⁺_¯z

(z, E), a

⁺_z¯

(z, E) = − 4∂

z

ln

Z

T

µ

+

(z, ζ, E) | dζ | ,

V

appr⁺

(z, E ) = 2i √ E∂

z¯

Z

T

µ

+

(z, ζ, E) dζ ζ

²

Z

T

µ

+

(z, ζ, E) dζ ζ

,

(3.19)

where

z

^{isgivenby(3.2).}

Step5. Find

A

^divappr,j

(x, E)

^,

V

appr^div

(x, E)

^,

x ∈ R

²,

j = 1

^,

2

^,

E > 0

^{,from formulas}

A

^divappr,1

(x, E) = i

8 (a

⁻_z

(z, E ) + a

⁺_¯z

(z, E)), A

^divappr,2

(x, E) = 1

8 (a

⁻_z

(z, E ) − a

⁺_¯z

(z, E)),

^(3.20)

V

appr^div

(x, E) = 1

2 V

appr⁻

(x, E) + V

appr⁺

(z, E)

− 1

8 a

⁻_z

(z, E)a

⁺_¯z

(z, E ),

where

z

^{isdenedby(3.2)andfuntions}

a

⁻_z^,

a

⁺z¯^,

V

appr^±

aredenedin(3.18),

(3.19).

Aderivationofthisreonstrutionalgorithmisbasedonthemethod ofthe

RiemannHilbert problem and on the

∂ ¯

^{-method. This derivation is given in}

Setion6.

For theasewhen

A ≡ 0

^{thisalgorithm isreduedtothealgorithmof[N4℄}

forapproximatiending

V

^on

R

²from

f

^on

M

E^{. Thealgorithmof[N4℄onsists}

ofthesameaforementionedsteps1,2,3andtheformula

V

^appr

= V

appr⁻ ,where

V

appr⁻

isdenedin (3.18). Thisalgorithmof[N4℄wasimplementednumerially

in[BAR℄.

Forthegeneralasethisalgorithmanbealsoregardedassimpliationand

development of the algorithm mentioned (in few lines) on page 457 of [N3℄.

Atually, in [N3℄ the part of the algorithm onsisting in nding

µ

_± ^from

h

_±

mentionedfortheasewhen

A

1

= A

1

, A

2

= A

2

, − 2i div A + V = V,

^(3.21)

i.e. for theself-adjointase, whereasthis assumption is not neessaryfor the

algorithm.

3.3. Properties ofthe algorithm. Let

k u

1

k

L²(T)

= Z

T

| u

1

(λ) |

²

| dλ |

1/2

,

k u

2

k

L²(T²)

= Z

T²

| u

2

(λ, λ

^′

) |

²

| dλ | | dλ

^′

|

1/2

, T

²

= T × T,

(3.22)

where

u

1^and

u

2^{aretestfuntions on}

T

^and

T

^2,respetively.

Proposition 3.1. Let

E > 0

^{be xed. Supposethat}

f ∈ L

²

(T

²

), k f k

L²(T²)

< 1

π ,

^(3.23)

where

f = f (λ, λ

^′

, E)

^{.Thenequation(3.12)isuniquelysolvablefor}

h

_±

∈ L

²

(T

²

)

and

k h

_±

k

L²(T²)

< k f k

L²(T²)

1 − π k f k

L²(T²)

,

^(3.24)

k B k

L²(T²)

< 2π k f k

L²(T²)

1 − π k f k

L²(T²)

,

^(3.25)

where

B

^{isdenedby(3.15), (3.16) (atxed}

z

^,

E

^{). In addition, if}

k f k

L²(T²)

< 1

3π ,

^(3.26)

then

k B k

L²(T²)

< 1

^{, equation (3.14), at xed}

z

^,

E

^{, is uniquely solvable for}

µ

⁺

∈ L

²

(T )

^and

k µ

⁺

k

L²(T)

< (2π)

^1/2

1 − k B k

L²(T²)

, k µ

⁺

− 1 k

L²(T)

< (2π)

^1/2

k B k

L²(T²)

1 − k B k

L²(T²)

,

^(3.27)

k µ

_±

− 1 k

L²(T)

< 3π(2π)

^1/2

k f k

L²(T²)

1 − 3π k f k

L²(T²)

,

^(3.28)

where

µ

_± ^{are denedby (3.17). Inaddition, atleast, if}

k f k

L²(T²)

< 1

6π ,

^(3.29)