On the reconstruction of conductivity of bordered two-dimensional surface in R^3 from electrical currents measurements on its boundary

On the reconstruction of conductivity of bordered two-dimensional surface in R ³

from electrical currents measurements on its boundary G.M. Henkin ¹ and R.G. Novikov ²

1 Universit´e Pierre et Marie Curie, case 247, 4 place Jussieu, 75252, Paris, France e-mail: henkin@math.jussieu.fr

2 CNRS (UMR 7641), Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, 91128, Palaiseau, France

e-mail: novikov@cmap.polytechnique.fr

Abstract.

An electrical potential U on a bordered real surface X in R ³ with isotropic conductivity function σ > 0 satisfies equation d(σd ^c U )

X = 0, where d ^c = i( ¯ ∂ − ∂), d = ¯ ∂ + ∂ are real operators associated with complex (conforme) structure on X induced by Euclidien metric of R ³ . This paper gives exact reconstruction of conductivity function σ on X from Dirichlet-to-Neumann mapping U

bX → σd ^c U

bX . This paper extends to the case of the Riemann surfaces the reconstruction schemes of R.Novikov [N2] and of A.Bukhgeim [B], given for the case X ⊂ R ² . The paper extends and corrects the statements of [HM], where the inverse boundary value problem on the Riemann surfaces was firstly considered.

Keywords. Riemann surface. Electrical current. Inverse conductivity problem. ¯ ∂- method.

Mathematics Subject Classification (2000) 32S65, 32V20, 35R05, 35R30, 58J32, 81U40.

0. Introduction

0.1. Reduction of inverse boundary value problem on a surface in R ³ to the corre- sponding problem on affine algebraic Riemann surface in C ³ .

Let X be bordered oriented two-dimensional manifold in R ³ . Manifold X is equiped by complex (conformal) structure induced by Euclidean metric of R ³ . We say that X possesses an isotropic conductivity function σ > 0, if any electric potential u on bX generates electrical potential U on X , solving the Dirichlet problem:

U

bX = u and dσd ^c U

X = 0, (0.1)

where d ^c = i( ¯ ∂ − ∂), d = ¯ ∂ +∂ and the Cauchy-Riemann operator ¯ ∂ corresponds to complex (conformal) structure on X. Inverse conductivity problem consists in the reconstruction of σ

X from the mapping potential U

bX → current j = σd ^c U

bX for solutions of (0.1).

This mapping is called Dirichlet-to-Neumann mapping.

This problem is the special case of the following more general inverse boundary value problem, going back to I.M.Gelfand [Ge] and A.Calderon [C]: to find potential (2-forme) q on X in the equation

dd ^c ψ = qψ (0.2)

from knowledge of Dirichlet-to-Neumann mapping ψ

bX → d ^c ψ

bX for solutions of (0.2).

Equation (0.2) is called in some context by stationary Schr¨ odinger equation, in other context by monochromatic acoustic equation etc. Equation (0.1) can be reduced to the equation (0.2) with

q = ^{dd √}

^√ _σ ^σ by the substitution ψ = √ σU .

Let restriction of Euclidean metric of R ³ on X have (in local coordinates) the form ds ² = Edx ² + 2F dxdy + Gdy ² = Adz ² + 2Bdzd¯ z + ¯ Ad¯ z ² ,

where z = x + iy, B = ^E+G ₄ , A = ^E−G−2iF ₄ . Put µ = ^{A ¯}

B+ √

B

−|A|

. By classical results (going back to Gauss and Riemann) one can construct holomorphic embedding ϕ : X → C ³ , using some solution of Beltrami equation: ¯ ∂ϕ = µ∂ϕ on X. Moreover, embedding ϕ can be chosen in such a way that ϕ(X) belongs to smooth algebraic curve V in C ³ . Using existence of embedding ϕ we can identify further X with ϕ(X).

0.2. Reconstruction schemes for the case X ⊂ R ² ≃ C .

For the case X = Ω ⊂ R ² the exact reconstruction scheme for formulated inverse problems was given in [N2], [N3] under some restriction (smallness assumption) for σ or q (see Corollary 2 of [N2]) . For the case of inverse conductivity problem, see (0.1), (0.2), when q = ^{dd √}

^√ _σ ^σ , restriction on σ in this scheme was eliminated by A.Nachman [Na] by the reduction to the equivalent question for the first order system studied by R.Beals and R.Coifman [BC2]. Recently A.Bukhgeim [B] has found new original reconstruction scheme for inverse boundary value problem, see (0.2), without smallness assumption on q.

In a particular case, the scheme of [N2] for the inverse conductivity problem consists in the following. Let σ(x) > 0 for x ∈ Ω and ¯ σ ∈ C ⁽²⁾ ( ¯ Ω). Put σ(x) = 1 for x ∈ R ² \ Ω. ¯

Let q = ^{dd √}

^√ _σ ^σ .

From L.Faddeev [F1] result it follows: ∃ compact set E ⊂ C such that for each λ ∈ C \ E there exists a unique solution ψ(z, λ) of the equation dd ^c ψ = qψ = ^{dd √}

^√ _σ ^σ ψ, with asymptotics

ψ(z, λ)e ^{−λz def} = µ(z, λ) = 1 + o(1), z → ∞ . Such solution can be found from the integral equation

µ(z, λ) = 1 + i 2

Z

ξ∈Ω

g(z − ξ, λ) µ(ξ, λ)dd ^c √

√ σ

σ , (0.3)

where the function g(z, λ) = i

(2π) ² Z

w∈ C

e ^{λw− λ ¯ w ¯} dw ∧ d w ¯

(w + z) ¯ w = i 2(2π) ²

Z

w∈ C

e ^{i(w z+ ¯ ¯ wz)} dw ∧ d w ¯ w( ¯ w − iλ) is called the Faddeev-Green function for the operator

µ 7→ ∂(∂ ¯ + λdz)µ.

From [N2] it follows that ∀ λ ∈ C \ E the function ψ

bΩ can be found through Dirichlet-to- Neumann mapping by integral equation

ψ(z, λ)

bΩ = e ^λz + Z

ξ∈bΩ

e ^λ(z−ξ) g(z − ξ, λ)( ˆ Φψ(ξ, λ) − Φ ˆ ₀ ψ(ξ, λ)), (0.4)

where ˆ Φψ = ¯ ∂ψ

bΩ , ˆ Φ ₀ ψ = ¯ ∂ψ ₀

bΩ , ψ ₀

bΩ = ψ

bΩ and ∂ ∂ψ ¯ ₀ Ω = 0.

By results of [BC1], [GN] and [N2] it follows that ψ(z, λ) satisfies ¯ ∂-equation of Bers- Vekua type with respect to λ ∈ C \ E :

∂ψ

∂ λ ¯ = b(λ) ¯ ψ, where (0.5)

λb(λ) = ¯ − 1 2πi

Z

z∈bΩ

e ^{λz− λ ¯ z ¯} ∂ ¯ _z µ(z, λ) = 1 4π

Z

Ω

e ^{λz− λ ¯ z ¯} qµ, (0.6) ψ(z, λ)e ^−λz = µ(z, λ) → 1, λ → ∞ , ∀ z ∈ C . (0.7) From [BC2] and [Na] it follows that for q = ^{dd √}

^√ _σ ^σ , σ > 0, σ ∈ C ⁽²⁾ ( ¯ Ω) the exceptional set E = {∅} and function λ 7→ b(λ) belongs to L ^2+ε ( C ) ∩ L ^2−ε ( C ) for some ε > 0. As a consequence function µ = e ^−λz ψ is a unique solution of the Fredholm integral equation

µ(z, λ) + 1 2π

Z

λ

∈ C

b(λ ^′ )e ^{¯ λ}

^{z−λ ¯}

^z µ(z, λ ¯ ^′ ) dλ ^′ ∧ d ¯ λ ^′

λ ^′ − λ = 1. (0.8)

Integral equations (0.4), (0.8) permit, starting from the Dirichlet-to-Neumann map- ping, to find firstly the boundary values ψ

bΩ , secondly ” ¯ ∂-scattering data” b(λ) and thirdly function ψ

Ω . From equality dd ^c ψ = ^{dd √}

^√ _σ ^σ ψ on X we find finally ^{dd √}

^√ _σ ^σ on X .

The scheme of the Bukhgeim type [B] can be presented in the following way. Let q = Qdd ^c | z | ² , where Q ∈ C ⁽¹⁾ ( ¯ Ω), but potential Q is not necessary of the conductivity form ^{dd √}

^√ _σ ^σ . By variation of Faddeev statement and proof we obtain that ∀ a ∈ C ∃ compact set E ⊂ C such that ∀ λ ∈ C \ E there exists a unique solution ψ a (z, λ) of the equation dd ^c ψ = qψ with asymptotics

ψ a (z, λ)e ^−λ(z−a)

= µ a (z, λ) = 1 + o(1), z → ∞ .

Such a solution can be found from integral equation (0.3), where kernel g(z, λ) is replaced by kernel

g _a (z, ζ, λ) = ie ^λa

^{− λ¯ ¯ a}

2π ²

Z

C

e ^{−λ(ζ−η+a)}

^{+¯ λ(¯ ζ− η+¯ ¯ a)}

(η − z)(¯ ζ − η) ¯ dη ∧ d¯ η.

Kernel g _a (z, ζ, λ) can be called the Faddeev type Green function for the operator

µ → ∂(∂ ¯ + λd(z − a) ² )µ. Equation ¯ ∂(∂ + λd(z − a) ² )µ = ₂ ⁱ qµ and Green formula implies Z

bΩ

e ^λ(z−a)

^{− λ(¯ ¯ z−¯ a)}

∂µ ¯ = Z

Ω

e ^λ(z−a)

^{− ¯ λ(¯ z−¯ a)}

qµ

2i . (0.9)

Stationary phase method, applied to the integral in the right-hand side of (0.9), gives for τ → ∞ , τ ∈ R , equality

τ→∞ lim

τ∈

R 4τ πi

Z

z∈bΩ

e ^iτ[(z−a)

^{+(¯ z−¯ a)}

^] ∂ ¯ _z µ _a (z, iτ ) = Q(a). (0.10)

Formula (0.10) means that values of potential Q in the arbitrary point a of Ω can be reconstructed from Dirichlet-to-Neumann mapping µ _a

bΩ 7→ ∂ ¯ _z µ _a

bΩ for family of functions µ _a (z, λ) depending on parameter λ = iτ , τ > const, where we assume that µ _a

bΩ is found using an analog of (0.4) for ψ a

bΩ .

Bukhgeim’s scheme works well at least ∀ Q ∈ C ⁽¹⁾ ( ¯ Ω).

More constructive scheme of [N2] works quite well only in the absence of exceptional set E in the λ-plane for Faddeev type functions. In papers [BLMP], [Ts], [N3] it was con- structed modified Faddeev-Green function permitting to solve inverse boundary problem (0.2), on the R ² = C , at least, under some smallness assumptions on potential Q.

Let us note that the first uniqueness results in the two-dimensional inverse boundary value or scattering problems for (0.1) or (0.2) goes back to A.Calderon [C], V.Druskin [D], R.Kohn, M.Vogelius [KV], J.Sylvester, G.Uhlmann [SU] and R.Novikov [N1].

Note in this connection that the first seminal results on reconstruction of the two- dimensional Schr¨ odinger operator H on the torus from the data ”extracted” from the family of eigenfunctions (Bloch-Floquet) of single energy level Hψ = Eψ were obtained in series of papers starting from B.Dubrovin, I.Krichever, S.P.Novikov [DKN], S.P.Novikov, A.Veselov [NV]. These results were obtained in connection with (2+1)- dimensional evo- lution equations.

This paper extends to the case of Riemann surfaces reconstruction procedures of [N2]

and of [B]. The paper extends (and also corrects) the recent paper [HM2] where the inverse boundary value problem on Riemann surface was firstly considered.Earlier in [HM1] it was proved that if X ⊂ R ³ possesses a constant conductivity then X with complex structure can be effectively reconstructed by at most three generic potential → current measurements on bX.

Very recently, motivated by [B] and [HM1], [HM2], C.Guillarmou and L.Tzou [GT]

have obtained general identifiability result (without reconstruction procedure): if for all solutions of equations dd ^c u + q _j u = 0, q _j ∈ C ⁽²⁾ (X ), j = 1, 2, Cauchy datas u

bX , d ^c u bX , coincide, then q ₁ = q ₂ on X .

1. Preliminaries and main results

Let C P ³ be complex projective space with homogeneous coordinates

w = (w 0 : w 1 : w 2 : w 3 ). Let C P _∞ ² = { w ∈ C P ³ : w 0 = 0 } . Then C P ³ \ C P _∞ ² can be considered as the complex affine space with coordinates z _k = w _k /w ₀ , k = 1, 2, 3. By classical result of G. Halphen (see R.Hartshorne [H], ch.IV, § 6) any compact Riemann surface of genus g can be embedded in C P ³ as projective algebraic curve ˜ V , which intersects C P _∞ ² transversally in d > g points, where d ≥ 1 if g = 0, d ≥ 3 if g = 1 and d ≥ g + 3 if g ≥ 2. Without loss of generality one can suppose that

i) V = ˜ V \ C P _∞ ² is connected affine algebraic curve in C ³ defined by polynomial equations V = { z ∈ C ³ : p ₁ (z) = p ₂ (z) = p ₃ (z) = 0 } such that the rang of the matrix [ ^∂p _∂z

(z), ^∂p _∂z

(z), ^∂p _∂z

(z)] ≡ 2 ∀ z ∈ V .

ii) ˜ V ∩ C P _∞ ² = { β ₁ , . . . , β _d } , where

β _l = (0 : β _l ¹ : β _l ² : β _l ³ ), β _l ² β _l ¹ , β _l ³

β _l ¹

∈ C ² , l = 1, 2, . . . , d.

iii) For r 0 > 0 large enough det

∂p

∂z

∂p

∂z

∂p

∂z

∂p

∂z

6 = 0 for z ∈ V : | z 1 | ≥ r 0 and α 6 = β.

iv) For | z | large enough:

dz ₂ dz 1

V

= γ _l + γ _l ⁰

z ₁ ² + O 1 z ₁ ³

, dz ₃ dz 1

V

= ˜ γ _l + ˜ γ _l ⁰

z ² ₁ + O 1 z ₁ ³

, where γ l , γ ˜ l , γ _l ⁰ , γ ˜ _l ⁰ 6 = 0, for l = 1, . . . , d, d ≥ 2.

Let V ₀ = { z ∈ V : | z ₁ | ≤ r ₀ } and V \ V ₀ = ∪ ^d l=1 V _l , where { V _l } are connected components of V \ V 0 . Let us equip V by Euclidean volume form dd ^c | z | ² . Let

W ˜ ^{1,˜ p} (V ) = { F ∈ L ^∞ (V ) : ¯ ∂F ∈ L ^p _0,1 ^˜ (V ) } , ˜ W _1,0 ^{1,˜ p} (V ) = { f ∈ L ^∞ _1,0 (V ) : ¯ ∂f ∈ L ^p _1,1 ^˜ (V ) } ,

˜

p > 2. Let H _0,1 (V ) denotes the space of antiholomorphic (0,1)-forms on V . Let H _0,1 ^p (V ) = H _0,1 (V ) ∩ L ^p _0,1 (V ), 1 < p < 2.

Let W ^1,p (V ) = { F ∈ L ^p (V ) : ¯ ∂F ∈ L ^p _0,1 (V ) } .

From the Hodge-Riemann decomposition theorem (see [GH], [Ho]) ∀ Φ ₀ ∈ W _0,1 ^1,p ( ˜ V ) we have

Φ 0 = ¯ ∂( ¯ ∂ ^∗ GΦ 0 ) + H Φ 0 , where H Φ 0 ∈ H 0,1 ( ˜ V ) and G is the Hodge-Green operator for the Laplacian ¯ ∂ ∂ ¯ ^∗ + ¯ ∂ ^∗ ∂ ¯ on ˜ V with the properties: G(H _0,1 ( ˜ V )) = 0, ¯ ∂G = G ∂, ¯ ¯ ∂ ^∗ G = G ∂ ¯ ^∗ .

Straight generalization of Proposition 1 from [He] gives explicit operators:

R ₁ : L ^p _0,1 (V ) → L ^{p ˜} (V ), R ₀ : L ^p _0,1 (V ) → W ˜ ^{1, p ˜} (V ) and H : L ^p _0,1 (V ) → H _0,1 ^p (V ), 1 < p < 2,

1

˜

p = ¹ _p − ¹ ₂ , such that ∀ Φ ∈ L ^p _0,1 (V ) we have decomposition of Hodge-Riemann type:

Φ = ¯ ∂RΦ + H Φ, where R = R ₁ + R ₀ , R ₁ Φ(z) = 1

2πi Z

ξ∈V

Φ(ξ) ∧ (dp _α ∧ dp _β ) ⌋ dξ ₁ ∧ dξ ₂ ∧ dξ ₃ det[ ∂p _α (ξ)

∂ξ , ∂p _β (ξ)

∂ξ , ξ ¯ − z ¯

| ξ − z | ² ], R 0 Φ(z) = ( ¯ ∂ ^∗ G( ¯ ∂R 1 Φ − Φ))(z) − ( ¯ ∂ ^∗ G( ¯ ∂R 1 Φ − Φ))(β 1 ),

( ¯ ∂R ₁ Φ − Φ) ∈ W _0,1 ^1,p ( ˜ V ), G is the Hodge − Green operator for Laplacian ¯ ∂ ∂ ¯ ^∗

for (0, 1) − forms on ˜ V ,

(1,1)-form under sign of integral does not depend on the choice of indexes α, β = 1, 2, 3, α 6 = β,

H Φ =

g

X

j=1

Z

V

Φ ∧ ω _j

¯ ω _j ,

{ ω j } is orthonormal basis of holomorphic (1,0)-forms on ˜ V , i.e.

Z

V

ω j ∧ ω ¯ k = δ jk , j, k = 1, 2, . . . , g.

Note that as a corollary of construction of R we have that lim

z∈V

RΦ(z) = RΦ(β ₁ ) = 0.

Remark 1.1. If V = { z ∈ C ² : P (z) = 0 } be algebraic curve in C ² then formula for operator R 1 is reduced to the following:

R 1 Φ(z) = 1 2πi

Z

ξ∈V

Φ(ξ) dξ 1

∂P

∂ξ

det ∂P

∂ξ (ξ), ξ ¯ − z ¯

| ξ − z | ² .

Remark 1.2. Based on [HP] one can construct an explicit formula not only for the main part R 1 of the R-operator, but for the whole operator R = R 1 + R 0 .

Let ϕ ∈ L ¹ _1,1 (V ) ∩ L ^∞ _1,1 (V ), f ∈ W ˜ _1,0 ^{1, p ˜} (V ), λ ∈ C , θ ∈ C . Let R ˆ θ ϕ = R((dz 1 + θdz 2 ) ⌋ ϕ)(dz 1 + θdz 2 ),

R λ,θ f = e _−λ,θ R(e λ,θ f), where e λ,θ (z) = e ^λ(z

^+θz

^{)− λ(¯ ¯ z}

^{+¯ θ¯ z}

⁾ .

By straight generalization of Propositions 2, 3 from [He] the form f = ˆ R _θ ϕ is a solution of ¯ ∂f = ϕ on V , function u = R λ,θ f is a solution of

(∂ + λ(dz 1 + θdz 2 ))u = f − H λ,θ f, where H λ,θ f ^def = e _−λ,θ H (e λ,θ f ), u ∈ W ^{1,˜ p} (V ), p > ˜ 2.

In addition, by straight generalization of Proposition 4 from [He] we have that

∂(∂ ¯ + λ(dz ₁ + θdz ₂ ))u = ϕ + ¯ λ(d¯ z ₁ + ¯ θd z ¯ ₂ ) ∧ H λ,θ ( ˆ R _θ ϕ) on V.

Definition 1.1. The kernel g _λ,θ (z, ξ), z, ξ ∈ V , λ ∈ C , of integral operator R _λ,θ ◦ R ˆ _θ is called in [He] the Faddeev type Green function for operator ¯ ∂(∂ + λ(dz 1 + θdz 2 )).

Definition 1.2. Let g = genus V ˜ . Let { ω _j } , j = 1, . . . , g , be orthonormal basis of holomorphic forms on ˜ V . Let { a 1 , . . . , a g } be different points (or effective divisor) on V \ V ₀ . Let

∆ θ (λ) = det Z

ξ∈V

R ˆ θ (δ(ξ, a j )) ∧ ω ¯ k (ξ)e λ,θ (ξ), j, k = 1, . . . , g,

where δ(ξ, a j )- Dirac (1,1)-form concentrated in { a j } . Let E _θ = { λ ∈ C : ∆ _θ (λ) = 0 } .

Definition 1.3. Parameter θ ∈ C will be called generic if θ / ∈ { θ ₁ , . . . , θ _d } , where θ _l = − 1/γ _l . Divisor { a ₁ , . . . , a _g } on V \ V ₀ will be called generic if

det ω _j dz ₁ (a k )

j,k=1,...,g 6 = 0.

Proposition 1.1. Let parameter θ ∈ C and divisor { a ₁ , . . . , a _g } on V \ V ₀ be generic, where V ₀ = { z ∈ V : | z ₁ | ≤ r ₀ } , g ≥ 1. Then for r ₀ large enough we have inequalities:

lim _λ→∞ | λ ^g ∆ θ (λ) | < ∞ and

∀ ε > 0 lim _λ→∞ | λ ^g ∆ θ (λ) | ε > 0, where | λ ^g ∆ θ (λ) | ε = sup

{λ

:|λ

−λ|<ε} | (λ ^′ ) ^g · ∆ θ (λ ^′ ) | Besides, the set E _θ is a closed nowhere dense subset of C .

Let X be domain containing V 0 and relatively compact on V . Let σ ∈ C ⁽³⁾ (V ), σ > 0, on V , σ = 1 on V \ X. Let Y be domain containing ¯ X and relatively compact on V . Let divisor { a 1 , . . . , a g } on Y \ X and parameter θ ∈ C be generic.

Definition 1.4. The functions ψ _θ (z, λ) = √

σF _θ (z, λ) = µ _θ (z, λ)e ^λ(z

^+θz

⁾ , z ∈ V , θ ∈ C \{ θ ₁ , . . . , θ _d } , λ ∈ C \ E _θ , will be called the Faddeev type functions, associated with σ, θ and { a ₁ , . . . , a _g } if ψ _θ , F _θ , µ _θ satisfy correspondingly properties:

dσd ^c F _θ = 2 √

σe ^λ(z

^+θz

⁾

g

X

j=1

C _j,θ (λ)δ(z, a _j ),

dd ^c ψ _θ = qψ _θ + 2e ^λ(z

^+θz

⁾

g

X

j=1

C _j,θ (λ)δ(z, a _j ),

∂(∂ ¯ + λ(dz ₁ + θdz ₂ ))µ _θ = i

2 qµ _θ + i

g

X

j=1

C _j,θ (λ)δ(z, a _j ),

(1.1)

and the normalization condition

z∈V lim

µ _θ (z, λ) = 1, (1.2)

where µ θ

Y ∈ L ^{p ˜} (Y ), µ θ

V \Y ∈ L ^∞ (V \ Y ), ˜ p > 2, q = ^{dd √}

^√ _σ ^σ , { C j,θ } are some functions of λ ∈ C \ E _θ .

Theorem 1.1. Under the aforementioned notations and conditions, ∀ generic θ ∈ C ,

∀ generic divisor { a ₁ , . . . , a _g } ⊂ V \ X and ∀ λ ∈ C \ E _θ : | λ | > const(V, { a _j } , θ, σ) there exists unique Faddeev type function

ψ _θ (z, λ) = √

σF _θ (z, λ) = e ^λ(z

^+θz

⁾ µ _θ (z, λ),

associated with conductivity function σ and divisor { a 1 , . . . , a g } . Moreover:

A) function z → ψ _θ (z, λ) and parameters { C _j,θ (λ) } can be found from the following equations, depending on parameters θ ∈ C , λ ∈ C \ E θ ,

ψ _θ (z, λ) − i 2

Z

ξ∈X

e ^λ((z

^−ξ

^)+θ(z

^−ξ

⁾⁾ g _λ,θ (z, ξ) dd ^c √

√ σ

σ ψ _θ (z, λ) = e ^λ(z

^+θz

⁾ + i

g

X

j=1

C j,θ (λ)g λ,θ (z, a j )e ^λ(z

^+θz

⁾ ,

(1.3)

2

g

X

j=1

C _j,θ (λ)e _λ,θ (a _j ) ω ¯ _k

d¯ z ₁ + ¯ θd¯ z ₂ (a _j ) =

− Z

z∈V

e ^{− λ(¯ ¯ z}

^{+¯ θ¯ z}

⁾ dd ^c √

√ σ

σ ψ _θ (z, λ) ω ¯ _k

d z ¯ ₁ + ¯ θd¯ z ₂ (z),

(1.4)

where k = 1, 2, . . . , g and { ω _j } is orthonormal basis of holomorphic forms on V ˜ ;

B) functions z → ψ _θ (z, λ) and parameters { C _j,θ (λ) } satisfy the following properties for λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ)

∃ lim

z→∞, z∈V

¯

z 1 + ¯ θ z ¯ 2

λ ¯ e ^{− ¯ λ(¯ z}

^{+¯ θ¯ z}

⁾ ∂ψ θ

∂ z ¯ ₁ + ¯ θ ∂ψ θ

∂¯ z ₂

= lim

z→∞

ψ _θ e ^−λ(z

^+θz

⁾ b _θ (λ), (1.5) iC _j,θ (λ) = (2πi)Res _a

e ^−λ(z

^+θz

⁾ ∂ψ _θ ^def = 2πi lim

ε→0

Z

|z−a

|=ε

e ^−λ(z

^+θz

⁾ ∂ψ _θ , (1.6)

∂ψ _θ (z, λ)

∂ λ ¯ = b _θ (λ)ψ _θ (z, λ), (1.7)

∂C j,θ (λ)

∂ λ ¯ e ^λ(a

^+θa

⁾ = b _θ (λ)C _j,θ (λ)e ^{¯ λ(¯ a}

^{+¯ θ¯ a}

⁾ . (1.8) Besides,

λb ¯ _θ (λ)d = − 1 2πi

Z

z∈bX

e _λ,θ (z) ¯ ∂µ(z) + i

g

X

j=1

C _j,θ e _λ,θ (a _j ),

| λ | · | b θ (λ) | ≤ const(V, { a j } , σ ) 1 ( | λ | + 1) ^1/3

1

| ∆ _θ (λ) | (1 + | λ | ) ^g ,

| C j,θ (λ) | ≤ const(V, { a j } , σ) 1 ( | λ | + 1) ^1/3

1

| ∆ _θ (λ) | (1 + | λ | ) ^g .

(1.9)

Remark 1.3. If k ln √

σ k C

(X) ≤ const(V, { a _j } , θ) then the condition

λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ) in Theorem 1.1 can be replaced by the condi- tion λ ∈ C \ E _θ . Dependence of const(V, { a _j } , θ, σ ) of σ means its dependence only of k ln √

σ k C

(X) .

Definition 1.5. The functions b θ (λ) and { C j,θ } will be called ”scattering” data for potential q.

Let ˆ Φ(ψ

bX ) = ¯ ∂ψ

bX for all sufficiently regular solutions ψ of (0.2) in ¯ X, where q = ^{dd √}

^√ _σ ^σ . The operator Φ is equivalent to the Dirichlet-to-Neumann operator for (0.1).

Let ˆ Φ ₀ denote ˆ Φ for q ≡ 0 on ¯ X.

Theorem 1.2. Under the conditions of Proposition 1.1 and Theorem 1.1, the follow- ing statements are valid:

A) ∀ λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ) the restriction of ψ _θ (z, λ) on bX and data { C _j,θ (λ) } can be reconstructed from Dirichlet-to-Neumann data as unique solution of the Fredholm integral equation

ψ _θ (z, λ) | bX + Z

ξ∈bX

e ^λ[(z

^−ξ

^)+θ(z

^−ξ

^)] g _λ,θ (z, ξ)( ˆ Φ − Φ ˆ ₀ )ψ _θ (ξ, λ) = (1.10)

e ^λ(z

^+θz

⁾ + i

g

X

j=1

C _j,θ (λ)g _λ,θ (z, a _j )e ^λ(z

^+θz

⁾ , where Z

z∈bX

(z ₁ + θz ₂ ) ^−k (∂ + λ(dz ₁ + θdz ₂ ))µ _θ (z, λ) = −

g

X

j=1

(a _j,1 + θa _j,2 ) ^−k C _j,θ (λ), (1.11) k = 2, . . . , g + 1, where (without restriction of generality) we suppose that values { a _j,1 } of the first coordinates of points { a _j } are mutually different;

B) Function σ(w), w ∈ X, can be reconstructed from Dirichlet-to- Neumann data ψ θ

bX

def = µ θ

bX e ^λ(z

^+θz

⁾ → ∂ψ ¯ θ

bX

by explicit formulas, where we assume that ψ _θ

bX is found using (1.10), (1.11).

For the case V = { z ∈ C ² : P (z) = 0 } , where P is a polynomial of degree N , this formula has the following form. Let { w m } be points of V , where (dz 1 + θdz 2 )

V (w m ) = 0, m = 1, . . . , M . Then for almost all θ values ^{√ dd} _σdd

^√

|z| ^σ

V (w _m ) can be found from the following linear system

τ (1 + o(1)) d ^k dτ ^k

Z

z∈bX

e _iτ,θ (z) ¯ ∂µ _θ (z, iτ )

=

M

X

m=1

iπ(1 + | θ | ² ) 2

dd ^c √

√ σ

σdd ^c | z | ² V

(w _m ) ×

| _∂z ^∂P

(w) | ³ _dτ ^d

exp iτ [(w _m,1 + θw _m,2 ) + ( ¯ w _m,n + ¯ θ w ¯ _m,2 )]

^∂

2

P

∂z

∂P

∂z

2

− 2 _∂z ^∂

^P

1

∂z

∂P

∂z

_∂P

∂z

+ ^∂ _∂z

^P

∂P

∂z

2

(w _m ) ,

(1.12)

where m, k = 1, . . . , M ; M = N (N − 1), τ ∈ R , τ → ∞ , | τ | ^g | ∆ _θ (iτ ) | ≥ ε > 0, ε- small

enough. Determinant of system (1.12) is proportional to the determinant of Vandermonde.

C) If g = 0 and if θ = θ(λ) = λ ⁻² , then ∀ z ∈ X and ∀ λ ∈ C function µ θ (z, λ) = ψ _θ (z, λ)e ^−λ(z

^+θz

is unique solution of Fredholm integral equation

µ _θ(λ) (z, λ) + 1 2πi

Z

ξ∈ C b _θ(ξ) (ξ)e ^{ξ(¯ ¯ z}

^{+¯ θ(ξ)¯ z}

^)−ξ(z

^+θ(ξ)z

⁾ µ ¯ _θ(ξ) (z, ξ) dξ ∧ d ξ ¯ ξ − λ = 1, where | b _θ(ξ) (ξ) | ≤ const(V )

(1 + | ξ | ) ² ,

and function z → σ(z), z ∈ X , can be found from equality dd ^c ψ _θ(λ) (z, λ) = dd ^c √

√ σ

σ (z)ψ _θ(λ) (z, λ), z ∈ X.

Remark 1.4. Using the Faddeev type Green function constructed in [He], in [HM2]

were obtained natural analogues of the main steps of the reconstruction scheme of [N2]

on the Riemann surface V . In particular, under a smallness assumption on ∂ log √ σ the existence (and uniqueness) of the solution µ(z, λ) of the Faddeev type integral equation

µ θ (z, λ) = 1 + i 2

Z

ξ∈V

g λ,θ (z, ξ) µ θ (ξ, λ)dd ^c √

√ σ

σ + i

g

X

j=1

C j g λ,θ (z, a j ), z ∈ V, λ ∈ C ,

holds for any a priori fixed constants C ₁ , . . . , C _g . However (and this fact was overlooked in [HM]) for λ ∈ C \ E there exists unique choise of constants C j (λ, σ) for which the integral equation above is equivalent to the differential equation

∂(∂ ¯ + λ(dz ₁ + θdz ₂ ))µ − i 2

dd ^c √

√ σ σ µ

+ i

g

X

j=1

C _j δ(z, a _j ),

where δ(z, a _j ) are Dirac measures concentrated in the points a _j .

2. Faddeev type functions on Riemann surfaces. Uniqueness Let projective algebraic curve ˜ V be embedded in C P ³ and intersect C P _∞ ² = { w ∈ C P ³ : w ₀ = 0 } transversally in d > g points. Let V = ˜ V \ C P _∞ ² , V ₀ = { z ∈ V : | z ₁ | ≤ r ₀ } and properties i)-iv) from § 1 be valid.

Proposition 2.1. Let σ be positive function belonging to C ⁽²⁾ (V ) such that σ ≡ const = 1 on V \ X ⊂ V \ V ₀ = ∪ ^d _l=1 V _l , where { V _l } are connected components of V \ V ¯ ₀ . Put q = ^{dd √}

^√ _σ ^σ . Let { a ₁ , . . . , a _g } be generic divisor with support in Y \ X, ¯ X ¯ ⊂ Y ⊂ Y ¯ ⊂ V . Let for generic θ ∈ C and λ ∈ C : | λ | ≥ const(V, { a _j } , θ, σ) function z 7→ µ = µ _θ (z, λ) be such that:

µ

Y ∈ L ^{p ˜} (Y ), µ

V \Y ∈ L ^∞ (V \ Y ¯ ),

∂µ ¯

Y ∈ L ^p (Y ), ∂µ ¯

V \ Y ¯ ∈ L ^{p ˜} (V \ Y ), 1 ≤ p < 2, p > ˜ 2, (2.1)

∂(∂ ¯ + λ(dz ₁ + θdz ₂ )µ = i

2 qµ + i

g

X

j=1

C _j δ(z, a _j ) with some C _j = C _j,θ (λ) and (2.2)

µ _θ (z, λ) → 0, z → ∞ , z ∈ V ₁ . (2.3)

Then µ _θ (z, λ) ≡ 0, z ∈ V .

Remark 2.1. Proposition 2.1 is a corrected version of Proposition 2.1 of [HM2]. For the case V = C the equivalent result goes back to [BC2].

Lemma 2.1. Let ψ = √

σF = e ^λ(z

^+θz

⁾ µ, where µ satisfies (2.1), (2.2) and F ₁ = √

σ∂F, F ₂ = √

σ ∂F. ¯ (2.4)

Then forms F ₁ , F ₂ satisfy the system of equations

∂F ¯ ₁ + F ₂ ∧ ∂ ln √

σ = ie ^λ(z

^+θz

⁾

g

X

j=1

C _j δ(z, a _j ),

∂F ₂ + F ₁ ∧ ∂ ¯ ln √

σ = − ie ^λ(z

^+θz

⁾

g

X

j=1

C _j δ(z, a _j ).

(2.5)

Proof of Lemma 2.1. From definition of F 1 and F 2 it follows that dσd ^c F = i[2σ∂ ∂F ¯ − ∂σ ¯ ∧ ∂F + ∂σ ∧ ∂F ¯ ] =

2i √

σ(∂F ₂ + F ₁ ∧ ∂ ¯ ln √

σ) = − 2i √

σ( ¯ ∂F ₁ + F ₂ ∧ ∂ ln √ σ).

From (2.4) and (2.2) we deduce also that d(σd ^c F ) = √

σ dd ^c ψ − ψ dd ^c √

√ σ σ

= 2 √

σe ^λ(z

^+θz

⁾

g

X

j=1

C _j δ(z, a _j ).

These equalities imply (2.5).

Lemma 2.1 is proved.

Lemma 2.2. Let { b _m } be the points of X, where (dz ₁ + θdz ₂ )

X (b _m ) = 0.

Let B ⁰ = ∪ m { b m } and A ⁰ = ∪ j { a j } .

Let u _± = m ₁ ± e _−λ,θ (z) ¯ m ₂ , where m ₁ = e ^−λ(z

^+θz

⁾ f ₁ , m ₂ = e ^−λ(z

^+θz

⁾ f ₂ , f ₁ = √

σ _∂z ^∂F

1

, f ₂ = √ σ _∂¯ ^∂F _z

1

. Let also q ₁ = ^{∂ ln} _∂z ^{√ σ}

1

and δ ₀ (z, a _j ) = _dz ^δ(z,a

⁾

1

∧d¯ z

. Then in conditions of Lemma 2.1

sup

z∈X | ∂u ¯ _±

X (z) · dist ² (z, B ⁰ ) | = O sup

z∈X | u _± dist(z, B ⁰ ) |

< ∞ ; u _±

V \X ∈ L ¹ (V \ X) ∩ O(V \ (X ∪ A ⁰ ))

(2.6)

and system (2.5) is equivalent to the system

∂u _±

∂ z ¯ 1

d z ¯ ₁ = ∓ (e _−λ,θ (z)q ₁ u ¯ _± )d¯ z ₁ + i

g

X

j=1

(C _j ± C ¯ _j e _−λ,θ (z))δ ₀ (z, a _j )d z ¯ ₁ .

(2.7)

Proof of Lemma 2.2. From (2.1) we deduce the property u _±

Y ∈ L ^p (Y ), 1 ≤ p < 2, u _±

V \Y ∈ L ^{p ˜} (V \ Y ) ⊕ L ^∞ (V \ Y ), p > ˜ 2.

System (2.5) is equivalent to the system of equations

∂f 1

∂ z ¯ ₁ = − f 2 q 1 + ie ^λ(z

^+θz

⁾

g

X

j=1

C j δ 0 (z, a j ),

∂f 2

∂z ₁ = − f 1 q ¯ 1 + ie ^λ(z

^+θz

⁾

g

X

j=1

C j δ 0 (z, a j ).

This system and definition of m ₁ , m ₂ imply

∂m ₁

∂ z ¯ 1

= − q ₁ m ₂ + i

g

X

j=1

C _j δ ₀ (z, a _j ),

∂m ₂

∂z 1

+ λm ₂ 1 + θ ∂z ₂

∂z 1

= − q ¯ ₁ m ₁ + i

g

X

j=1

C _j δ ₀ (z, a _j ).

From the last equalities and definition of u _± we deduce

∂u _±

∂ z ¯ ₁ = ∂m 1

∂ z ¯ ₁ ± e _−λ,θ (z) ∂ m ¯ 2

∂ z ¯ ₁ + ¯ λ 1 + ¯ θ ∂¯ z 2

∂¯ z ₁ m ¯ ₂

= − q ₁ m ₂ + i

g

X

j=1

C _j δ ₀ (z, a _j ) ±

e _−λ,θ (z) ¯ λ 1 + ¯ θ ∂ z ¯ 2

∂ z ¯ ₁

m ¯ ₂ − λ ¯ m ¯ ₂ 1 + ¯ θ ∂¯ z 2

∂¯ z ₁

− q ₁ m ¯ ₁ + i

g

X

j=1

C ¯ _j δ ₀ (z, a _j )

=

∓ (e _−λ,θ (z)q ₁ u ¯ _± ) + i

g

X

j=1

(C _j ± C ¯ _j e _−λ,θ (z))δ ₀ (z, a _j ).

Property (2.7) is proved.

For proving (2.6) we will use construction coming back to Bers and Vekua (see [Ro], [V]). Let β _± be continuous on Y solutions of ¯ ∂- equations

∂β ¯ _± = ± e _−λ,θ (z)q 1

¯

u _±

u _± d z ¯ 1 ,

where the right-hand side belongs to L ^∞ _0,1 (Y ).

Functions v _± = u _± e ^−β

belongs to O (Y ). Indeed, from (2.1), (2.2) it follows that µ ∈ W ^1,p (Y ) ∩ W _loc ^{1, p ˜} (Y \ (A ⁰ ∪ B ⁰ )). From this and from definition of v _± we deduce that

∂v ¯ _± = q 1 u ¯ _± d¯ z 1 e ^−β

− q 1 u _± ^u _u ^¯

e ^−β

d z ¯ 1 = 0 on Y \ (A ⁰ ∪ B ⁰ ) and the following formula for u _± is valid

u _± (z) = v _± (z)e ^β

^(z) . (2.8)

From this and (2.7), (2.8) we obtain (2.6).

Lemma 2.2 is proved.

Lemma 2.3. Let u _± be the functions from Lemma 2.2 and µ be the function from Lemma 2.1. Then

u _± = ∂µ

∂z 1

+ λ 1 + θ ∂z ₂

∂z 1

µ − q ₁ µ ± e _−λ,θ (z) ∂¯ µ

∂z 1 − q ₁ µ ¯ .

Proof of Lemma 2.3. We have

u _± = e ^−λ(z

^+θz

⁾ f ₁ ± e ^−λ(z

^+θz

⁾ f ¯ ₂ = e ^−λ(z

^+θz

⁾ (f ₁ ± f ¯ ₂ ), where

f ₁ = √ σ ∂F

∂z ₁ = √ σ ∂

∂z ₁

√ 1

σ e ^λ(z

^+θz

⁾ µ

= e ^λ(z

^+θz

⁾ ∂µ

∂z 1

+ λ 1 + θ ∂z ₂

∂z 1

µ − q ₁ µ , f ¯ ₂ = √

σ ∂ F ¯

∂z 1

= √ σ ∂

∂z 1

√ 1

σ e ^{λ(¯ ¯ z}

^{+¯ θ¯ z}

⁾ µ ¯

= e ^{λ(¯ ¯ z}

^{+¯ θ¯ z}

⁾ ∂ µ ¯

∂z ₁ − q 1 µ ¯ . This imply Lemma 2.3.

Lemma 2.4. Let ω ₁ , . . . , ω _g be orthonormal basis of holomorphic 1-forms on V ˜ . Let { a ₁ , . . . , a _g } be generic divisor on Y \ X ¯ , where V ₀ ⊂ X ¯ ⊂ Y ⊂ V . Put ω _j,k ⁰ = _dz ^ω

1

(a _j ). Let for some generic θ ∈ C and λ ∈ C functions u _± from Lemmas 2.2-2.3 satisfy (2.6), (2.7) with some C _j = C _j,θ (λ). Then

sup

j | C j,θ (λ) | ≤ const(V, { a j } , θ) k ln √

σ k ² _W

_(X) (1 + | λ | ) ^−1/3 k u _± k L

(X,B

) , where k u _± k L

(X,B

)

def = sup

z∈X | u _± (z)dist(z, B ⁰ ) | .

Proof of Lemma 2.4. From condition iv) of section 1 we deduce | ω ⁰ _j,k | < ∞ . From

definition of generic divisor we obtain det[ω _j,k ⁰ ] 6 = 0. From (2.7) and from definition of

Dirac measure ∀ k = 1, . . . , g we deduce

r→∞ lim

Z

{z∈V : |z

|=r}

u _± ∧ ω k

± Z

X

e _−λ,θ (z) ∂ ln √ σ

∂z 1 u ¯ _± d z ¯ 1 ∧ ω k = i

Z

Y g

X

j=1

(C _j ± C ¯ _j e _−λ,θ (z))δ ₀ (z, a _j )d z ¯ ₁ ∧ ω _k =

i

g

X

j=1

(C _j ± C ¯ _j e _−λ,θ (a _j ))ω _j,k ⁰ , j, k = 1, 2, . . . , g.

(2.9)

From estimates lim

r

→∞ sup

{z∈V : |z

|=r

} | u _± (z) | < ∞ , for some sequence r _n → ∞ , and

|ω

|

dz

≤ O | _z ¹

1

|

, z ∈ V \ Y , k = 1, . . . , g , we obtain

r→∞ lim

Z

{z∈V : |z

|=r}

u _± ∧ ω _k

= 0. (2.10)

From (2.9), (2.10) and Kramers’s formula we obtain i(C _j ± C ¯ _j e _−λ,θ (a _j )) =

det[ω _1,k ⁰ ; . . . ; ω _j−1,k ⁰ ; R

X ± e _−λ,θ (z) ^{∂ ln} _∂z ^{√ σ}

1

u ¯ _± d z ¯ ₁ ∧ ω _k ; ω _j+1,k ⁰ ; . . . ; ω _g,k ⁰ ]

det[ω _j,k ⁰ ] ,

(2.11)

where j, k = 1, . . . , g.

Let us prove estimate

Z

X

e _−λ,θ (z) ∂ ln √ σ

∂z ₁ u ¯ _± d¯ z ₁ ∧ ω _k ≤ const(X, θ)(1 + | λ | ) ^−1/3 k ln √

σ k ² _W

_(X) · k u _± k L

(X,B

) .

(2.12)

For | λ | ≤ 1 estimate follows directly, using that ln √

σ ∈ W ^1,∞ (X).

Let B ^ε = ∪ ^M m=1 { z ∈ X : | z − b m | ≤ ε } .

Let χ _ε,ν , ν = 1, 2, be functions from C ⁽¹⁾ (V ) such that χ _ε,1 + χ _ε,2 ≡ 1 on V , supp χ _ε,1 ⊂ B ^2ε , supp χ _ε,2 ⊂ V \ B ^ε , | dχ _ε,ν | = O( ¹ _ε ), ν = 1, 2.

Put J _ν ^ε u _± = R

X

χ _ε,ν (z)e _−λ,θ (z) ^{∂ ln} _∂z ^{√ σ}

1

u ¯ _± d¯ z ₁ ∧ ω _k , ν = 1, 2. We have directly:

| J ₁ ^ε u _± | ≤ const(X)ε k ln √

σ k W

(X) · k u _± k L

(X,B

) . (2.13)

For J ₂ ^ε u _± we obtain by integration by parts:

J ₂ ^ε u _± = − 1 λ

Z

X

χ _ε,2 ∂e _−λ,θ (z) ∂ ln √ σ

∂z 1

¯

u _± d¯ z ₁ ∧ ω _k dz 1 + θdz 2

= 1

λ Z

X

e _−λ,θ (z)∂

χ _ε,2 ∂ ln √ σ

∂z ₁ u ¯ _± d z ¯ ₁ ∧ ω k

dz ₁ + θdz ₂

.

(2.14)

To estimate (2.14) we use (2.6) and the following properties: | ∂χ ε,2 | = O( ¹ _ε ), supp(∂χ _ε,2 ) ⊂ B ^2ε ,

k ∂ ln √ σ

∂z 1

d z ¯ ₁ ∧ ∂χ _ε,2 u _± ω _k

dz 1 + θdz 2 k L

(X) ≤ const(X, θ)

ε k ln √

σ k W

(X) k u _± k L

(X,B

)

k ∂ ² ln √ σ

∂z ₁ ² dz ₁ ∧ d¯ z ₁ χ _ε,2 u _± ω _k

dz 1 + θdz 2 k L

(X) ≤

| ln ε | const(X, θ) k ln √

σ k W

(X) k u _± k L

(X,B

)

k ∂ ln √ σ

∂z ₁ d z ¯ ₁ χ _ε,2 u _± ∧ ∂ ω k

dz ₁ + θdz ₂

k L

(X) ≤ const(X, θ)

ε k ln √

σ k W

(X) k u _± k L

(X,B

)

∂ u ¯ _±

X = ∓ (e _λ,θ (z)¯ q ₁ u ¯ _± )dz ₁ . From (2.14), (2.6) and these properties we obtain

| J ₂ ^ε u _± | ≤ | ln ε | const(X, θ)

| λ | k ln √

σ k W

(X) · k u _± k L

(X,B

) + const(X, θ)

ε | λ | k ln √

σ k W

(X) · k u _± k L

(X,B

) + const(X, θ, δ)

ε ^1+δ | λ | k ln √

σ k W

(X) · k u _± k L

(X,B

) .

(2.15)

Putting in (2.13), (2.15) ε = ^{√ 1}

λ and δ = 1/3 we obtain (2.12) for | λ | ≥ 1.

Inequalities (2.11), (2.12) imply estimate

| C j ± C ¯ j e _−λ,θ (a j ) | ≤ const(X, { a j } , θ)(1 + | λ | ) ^−1/3 k ln √

σ k ² _W

_(X) · k u _± k L

(X,B

) . We obtained statement of Lemma 2.4.

Lemma 2.5. Let functions u _± satisfy (2.6), (2.7) and R - operator from section 1.

Then

k R[e _−λ,θ q ₁ u ¯ _± d ξ ¯ ₁ k L

(X,B

) ≤ const(X, θ)(1 + | λ | ) ^−1/5 k ln √

σ k W

(X) · k u _± k L

(X,B

) .

Proof of Lemma 2.5.

Let χ _ε,ν , ν = 1, 2, be partition of unity from Lemma 2.4. Put S _ν ^ε u _± = R[χ _ε,ν q ₁ u ¯ _± d ξ ¯ ₁ ], ν = 1, 2. Using (2.6) and formula for operator R we deduce estimate

k S ₁ ^ε u _± k L

(X,B

) = O(ε) k ln √

σ k W

(X) k u _± k L

(X,B

) . (2.16) Let R _1,0 (ξ, z) be kernel of operator R. It means, in particular, that ¯ ∂ _ξ R _1,0 (ξ, z) = − δ(ξ, z), where δ(ξ, z)- Dirac (1,1)- measure, concentrated in the point ξ = z. We have

S ₂ ^ε u _± = Z

X

χ _ε,2 e _−λ,θ q ₁ u ¯ _± d ξ ¯ ₁ R _1,0 (ξ, z). (2.17)

Integration by parts in (2.17) gives the following S ₂ ^ε u _± = 1

λ ¯ Z

X

∂e ¯ _−λ,θ (ξ) d ξ ¯ ₁

d ξ ¯ ₁ + ¯ θd ξ ¯ ₂ χ _ε,2 (ξ)q ₁ (ξ)¯ u _± (ξ)R _1,0 (ξ, z) =

− 1 λ ¯

Z

X

e _−λ,θ (ξ) ¯ ∂

d ξ ¯ ₁

d ξ ¯ ₁ + ¯ θd ξ ¯ ₂ χ _ε,2 (ξ)q ₁ (ξ)¯ u _± (ξ)

R _1,0 (ξ, z)+

1

λ ¯ e _−λ,θ (z) d ξ ¯ ₁

d ξ ¯ ₁ + ¯ θd ξ ¯ ₂ (z)χ _ε,2 (z)q ₁ (z)¯ u _± (z).

(2.18)

To estimate (2.18) we use (2.6), properties of partition of unity { χ _ε,ν } and inequalities

d ξ ¯ 1

d ξ ¯ 1 + ¯ θd ξ ¯ 2

(ξ)

= O 1

dist(ξ, B ⁰ ) ,

∂ ¯ d ξ ¯ 1

d ξ ¯ 1 + ¯ θd ξ ¯ 2

(ξ)

= O 1

(dist(ξ, B ⁰ )) ² ,

| q 1 (ξ) | = O 1 dist(ξ, B ⁰ )

, | ∂q ¯ 1 (ξ) | = O 1 (dist(ξ, B ⁰ )) ²

, ξ ∈ X.

(2.19)

From (2.19), (2.8) and from the formula for operator R we deduce estimate k S ₂ ^ε u _± k L

(X) = O( 1

ε ⁴ | λ | ) k ln √

σ k W

(X) k u _± k L

(X,B

) . (2.20) Putting in (2.16), (2.20) ε = _|λ| ¹

we obtain statement of Lemma 2.5.

Proof of Proposition 2.1.

Let function µ satisfy conditions (2.1)-(2.3) and u _± be functions defined in Lemma 2.2. Then by Lemma 2.3 we have

z→∞ lim

z∈V1

u _± (z, λ) = lim

z→∞

(m ₁ ± e _−λ,θ (z) ¯ m ₂ ) =

z→∞ lim

z∈V1

[λ 1 + θ dz ₂ dz ₁

µ + ∂µ

∂z ₁ ± e _−λ,θ (z) ∂ µ ¯

∂z ₁ ] → 0.

(2.21)

Let

h _± = u _± ± R[(e _−λ,θ (z)q ₁ u ¯ _± )d¯ z ₁ − i

g

X

j=1

(C _j ± C ¯ _j e _−λ,θ (z))δ ₀ (z, a _j )d z ¯ ₁ ], (2.22)

where R is the operator from section 1.

By Lemmas 2.2-2.5 and properties of operator R we have h _± ∈ O (V ) ∩ L ^∞ (V ) and h _± (z, λ) → 0, z → ∞ , z ∈ V ₁ . By Liouville theorem, h _± (z, λ) ≡ 0 on V , λ ∈ C . Then from (2.22) with h _± (z, λ) ≡ 0 and Lemmas 2.4, 2.5 it follows that u _± (z, λ) ≡ 0, z ∈ V , if λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ) k ln √

σ k ² _W

_(X) . Property u _± (z, λ) ≡ 0, z ∈ V , implies by Lemma 2.3 equality _∂¯ ^∂µ _z

1

− q ¯ ₁ µ = 0, z ∈ V , where µ(z) → ∞ if z ∈ V ₁ , z → ∞ . The Liouville type theorem for generalized holomorphic functions ([Ro], theorem 7.1) implies µ ≡ 0. Proposition 2.1 is proved.

3. Faddeev type functions on Riemann surface. Existence.

Proof of Theorem 1.1A

Proposition 3.1. Let conductivity σ and divisor { a ₁ , . . . , a _g } satisfy conditions of Proposition 2.1. Then ∀ generic θ ∈ C and ∀ λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ) there exists unique Faddeev type function

ψ ^def = √

σF ^def = e ^λ(z

^+θz

⁾ µ, where

ψ = ψ θ (z, λ), F = F θ (z, λ), µ = µ θ (z, λ), (3.1) associated with σ and divisor { a ₁ , . . . , a _g } , i.e.

∂(∂ ¯ + λ(dz 1 + θdz 2 ))µ = i 2 qµ +

g

X

j=1

C j δ(z, a j ), for some C j = C j,θ (λ), where q = dd ^c √

√ σ

σ , µ

Y ∈ L ^{p ˜} (Y ), µ

V \ Y ¯ ∈ L ^∞ (V \ Y ¯ ), lim

z→∞

µ θ (z, λ) = 1.

(3.1a)

In addition,

k µ _θ (z, λ) − µ _θ ( ∞ l , λ) k L

(V ) ≤ const(V, { a _j } , θ, σ, p, ε) ˜

| ∆ _θ (λ) | · (1 + | λ | ) ^g+1−ε , where µ _θ ( ∞ l , λ) ^def = lim

z→∞

µ _θ (z, λ), l = 1, . . . , d,

k ∂µ k L

(Y ) + k ∂µ k _L

_{(V \Y )} ≤ const(V, { a _j } , θ, σ, p, p, ε) ˜

| ∆ _θ (λ) | · (1 + | λ | ) ^g−ε , p < 2, p > ˜ 2,

(3.1b)

∀ generic θ ∈ C and λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ),

∂µ

∂ λ ¯

Y ∈ W ^1,p (Y ), ∂µ

∂ ¯ λ

V

\Y ∈ L ^∞ (V _l \ Y ) ∪ W ^{1, p ˜} (V _l \ Y ), (3.1c)

where { V l } are connected components of V \ V 0 , l = 1, . . . , d, e _λ,θ (z) = e ^λ(z

^+θz

^{)− λ(¯ ¯ z}

^{+¯ θ¯ z}

⁾ .

Remark 3.1. Proposition 3.1 is a corrected version of Proposition 2.2 from [HM2].

For the case V = C the results of such a type goes back to [F1], [F2].

Lemma 3.1. Under the conditions of Proposition 3.1, ∀ λ ∈ C \ E _θ function z → µ θ (z, λ) belonging to L ^{p ˜} (Y ) on Y and to L ^∞ (V \ Y ) on V \ Y satisfies (3.1a) iff

there exists C _j = C _j,θ (λ), j = 1, . . . , g , such that µ θ (z, λ) = 1 + i

2 Z

ξ∈X

g λ,θ (z, ξ)qµ θ (ξ, λ) + i

g

X

j=1

C j,θ (λ)g λ,θ (z, a j ) (3.2) and one of two equivalent conditions is valid

H λ,θ R ˆ θ i 2 qµ

+ i

g

X

j=1

C j,θ (λ) H λ,θ ( ˆ R θ (δ(z, a j )) = 0 or

(∂ + λ(dz 1 + θdz 2 ))µ θ (z, λ) ∈ H 1,0 (V \ (X ∪ ^g _j=1 { a j } )) ∩ L ¹ _1,0 (Y \ X),

(3.3)

where g _λ,θ is Faddeev type Green function, R ˆ _θ , H λ,θ - operators defined in section 1.

Proof of Lemma 3.1. From Proposition 4 in [He] and from definition of Green function g λ,θ (z, ξ) we deduce that integral equation (3.2) is equivalent to the following differential equation

∂(∂ ¯ + λ(dz ₁ + θdz ₂ ))µ = i

2 qµ + i

g

X

j=1

C _j,θ δ(z, a _j )+

λ(d¯ ¯ z ₁ + ¯ θd z ¯ ₂ )) ×

H λ,θ R ˆ _θ i 2 qµ

+ i

g

X

j=1

C _j,θ H λ,θ ( ˆ R _θ (δ(z, a _j )) .

(3.4)

Equation (3.4) is equivalent to (3.1a) if one of two equivalent conditions (3.3) is valid.

Lemma 3.1 is proved.

Lemma 3.2. Let { a ₁ , . . . , a _g } be generic divisor in Y \ X. Then for any generic ¯ θ ∈ C and

∀ λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ), integral equation (3.2), (3.3) is uniquely solvable Fredholm integral equation in the space W ˜ ^{1,˜ p} (V ).

Proof of Lemma 3.2. Let θ ∈ C and λ ∈ C \ E θ : | λ | ≥ const(V, { a j } , θ, σ). From (3.2), (3.3) we obtain integral equation for ˜ µ _θ = µ _θ − 1 and ˜ C _j,θ :

˜

µ _θ (z, λ) − i 2

Z

ξ∈V

g _λ,θ (z, ξ)q(ξ)˜ µ _θ (ξ, λ) − i

g

X

j=1

C ˜ _j,θ (λ)g _λ,θ (z, a _j ) =

i 2

Z

ξ∈V

g λ,θ (z, ξ)q(ξ) + i

g

X

j=1

C _j,θ ⁰ (λ)g λ,θ (z, a j ).

(3.5)

Parameters ˜ C j = ˜ C j,θ (λ), j = 1, . . . , g, are defined by the equations:

− i

g

X

j=1

C ˜ _j Z

V

R ˆ _θ (δ(ξ, a _j ))¯ ω _k (ξ)e _λ,θ (ξ) = Z

ξ∈V

e _λ,θ (ξ) ˆ R _θ i 2 q˜ µ

¯

ω _k (ξ), k = 1, 2, . . . , g.

(3.6)

We remind that determinant of system (3.6) is exactly ∆ θ (λ).

Parameters C _j,θ ⁰ are defined by (3.6) with C _j,θ ⁰ in place of ˜ C _j,θ and 1 in place of ˜ µ.

One can see also that C _j,θ ⁰ (λ) = C _j,θ (λ) − C ˜ _j,θ (λ).

Let us prove that (3.5), (3.6) determine Fredholm integral equation in the space W ˜ ^{1,˜ p} (V ), ˜ p > 2.

Propositions 2, 3 of [He] imply that correspondance

˜

µ 7→ R _λ,θ ◦ ( ˆ R _θ i 2 q µ ˜

+ i

g

X

j=1

C ˜ _j,θ R ˆ _θ (δ(z, a _j ))

define linear continuous mapping of ˜ W ^{1, p ˜} (V ) into itself. This mapping is compact because mapping ˜ µ → q˜ µ, supp q ⊂ X, from ˜ W ^{1, p ˜} (V ) into L ^p _1,1 ^˜ (X ) is compact, operator

R ˆ _θ : L ^p _1,1 ^˜ (X ) → W ˜ _1,0 ^{1, p ˜} (V ) and operator R _λ,θ : ˜ W _1,0 ^{1, p ˜} (V ) → W ˜ ^{1,˜ p} (V ) are bounded.

If for fixed λ 6∈ E _θ Fredholm equation (3.5), (3.6) is not solvable then correspond- ing homogeneous equation, when the right-hand side of (3.5) is replaced by zero, admits nontrivial solution ˜ µ ^∗ = µ ^∗ − 1.

By Lemma 3.1 function ˜ µ ^∗ satisfies differential equation (2.2) with C _j replaced by ˜ C _j and with property ˜ µ ^∗ (z) → 0, z → ∞ , z ∈ V ₁ .

By Proposition 2.1, ˜ µ ^∗ ≡ 0 if λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ).

It means that equation (3.2), (3.3) is uniquely solvable Fredholm integral equation for any λ ∈ C \ E _θ : | λ | ≥ const(V, { a _j } , θ, σ).

Lemma 3.2 is proved.

Lemma 3.3. Let { a ₁ , . . . , a _g } be generic divisor on Y \ X. Let λ ∈ C \ E _θ . Let µ be solution of integral equation (3.2), (3.3). Then relations (3.3) determining parameters C _j = C _j,θ (λ) are reduced to the following explicit formulas

2i

g

X

j=1

C _j,θ e _λ,θ (a _j ) ω ¯ _k d z ¯ 1

(a _j ) = Z

z∈X

e _λ,θ (z) i dd ^c √

√ σ

σ + 2 ¯ ∂ ln √

σ ∧ ∂ ln √ σ

µ ω ¯ _k d z ¯ 1

(z). (3.7)

Proof of Lemma 3.3. By Lemma 3.1 equations (3.2), (3.3) are equivalent to the equation:

∂(∂ ¯ + λ(dz 1 + θdz 2 ))µ = i

2 qµ + i

g

X

j=1

C j,θ δ(z, a j ), (3.8)