INVERSE SCATTERING FOR WAVEGUIDES

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Théorie spectrale et géométrie

Hiroshi ISOZAKI, Yaroslav KURYLEV& Matti LASSAS Inverse Scattering for Waveguides

Volume 25 (2006-2007), p. 71-83.

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INVERSE SCATTERING FOR WAVEGUIDES

Hiroshi Isozaki, Yaroslav Kurylev & Matti Lassas

Abstract. — We study the inverse scattering problem for a waveguide(M,g) with cylindrical ends,M =M^c∪ ∪^N_α=1(Ω^α×(0,∞))

, where eachΩ^α×(0,∞) has a product type metric. We prove, that the physical scattering matrix, measured on just one of these ends, determines(M,g)up to an isometry.

1. General formulation of the problem

One of the most typical geometric constructions encountered in the every- day life is that of a compound waveguide, e.g. setting of optical and elec- tric cables, oil, gas and water pipelines, etc. Mathematically, a compound waveguide can be represented as a non-compact connected Riemannian manifold (M,g)such that at infinity it looks like a disjoint union of the asymptotically cylindrical ends. More precisely, letX0be a point inM and denote byd(X, X₀)the distance betweenX₀and a variable pointX ∈M. Then, for large R > 0, the sphere SR(X0) = {X ∈ M : d(X, X0) =R}

decomposes into a finite number,N of the(n−1)−dimensional compact, connected submanifolds,(Ω^α_R, g_R^α),

S_R(X₀) =

N

[

α=1

Ω^α_R,

and, in a proper sense, (Ω^α_R, g_R^α)→ (Ω^α, g^α), when R → ∞. In this case, the Laplace operator, ∆_M in L²(M) (with either Dirichlet or Neumann boundary conditions on components of ∂M, if ∂M 6=∅) is defined as the closure of the Laplace operator on C^∞−functions with compact support.

It is then possible to develop a spectral and scattering theory for∆M, see e.g. [10], [11], [4], [13] and, in a more general setting, [19], [2], [3].

Math. classification:58J50, 35R30.

To describe the nature of the scattering theory, note that each asymptotic section(Ω^α, g^α), which we associate with the corresponding channel of scat- tering, gives rize to its own(n−1)−dimensional Laplacian,∆^α(for∂Ω^α6=∅ with the boundary conditions inherited from those on ∂M). Denote by {λ^α_m, φ^α_m}^∞_m=1 the eigenvalues and corresponding orthonormal eigenfunc- tions of∆^α. It is then possible to establish an existence, for all but a dis- crete number ofk²∈R+, of a generalized eigenfunctionψ^α_m(X;k²)of∆M, associated with{λ^α_m, φ^α_m}, whenλ^α_m< k². Physically, we can imagine the situation as follows: We send through the channel associated withΩ^α an incoming propagating wave intoM that is related to the eigenfunctionφ^α_m with λ^α_m < k². This is the wave which behaves, when d(X, X0) → ∞, asexp

−id(X, X0)p

k²−λ^α_m

φ^α_m(x^α)(for the terminology used see [22]).

Here X = (x^α, d(X, X0)), x^α being (local) coordinates in Ω^α, are (local) coordinates in the channelαwhich, for larged(X, X0), is diffeomorphic to Ω^α×(A,+∞)with some A >0. The described incoming wave propagates throughM, giving rize to the wavefunctionψ^α_m(X;k²), which is actually a generalised eigenfunction of the continuous spectrum for∆_M. In general, this wavefunction goes to infinity through allNchannels ofM. Registering ψ_m^α(X;k²)in theβ−channel asd(X, X0)→ ∞, we observe the asymptotic behaviour of the wave,

(1.1) ψ_m^α(X;k²)∼ X

λ^β_l<k²

S^αβ_ml(k²)e^iy

pk²−λ^β_lφ^β_l(x^β).

Herey =d(X, X₀)and X = (x^β, y), x^β ∈Rⁿ⁻¹ being (local) coordinates inΩ^β, are coordinates in theβ−channelΩ^β×(A,+∞). This construction is valid for all but a discrete set σ^s(M) of k², with σ^s(M) consisting of the (positive) eigenvalues of the point spectrum of −∆M and all k² = λ^α_m, α = 1, . . . , N;m = 1,2, . . . In the following, denote by d^β(k) the integer such that λ^β_l 6 k² for l 6 d^β(k) and λ^β_l > k² for l > d^β(k).

The above construction associates with any k² ∈ R+\ σ^s(M) a finite- dimensional matrix, calledthe physical scattering matrix,S(k²),

(1.2) S(k²) = [S^αβml(k²)]α,β∈{1,...,N}, m6d^α(k), l6d^β(k).

Note that the dimension of S(k²) is a step function with jumps at the eigenvalues of−∆^α, α∈ {1, . . . , N}.

A natural inverse scattering problem for a compound waveguide is to reconstruct, up to an isometry, the manifold(M,g)from its physical scat- tering matrixS(k²), k²∈R+\σ^s(M).Moreover, as it is quite conceivable that we can not make measurements in all N channels of M or even the

number of these channels isa prioriunknown, a more realistic inverse prob- lem isto reconstruct, up to an isometry, the manifold(M,g)from its partial physical scattering matrixS^αβml(k²), known for α, β∈K⊂ {1, . . . , N} and k²∈O, whereO is an infinite open subset ofR+such thatO∩σ^s(M) =∅.

Note that, making measurements on a part of the boundary at infinity of M, namely Ω^α, α ∈ K, it is natural to assume that the Riemannian manifolds(Ω^α, g^α)and, therefore, the pairs{λ^α_m, φ^α_m}, α∈K, m= 1,2. . . , of eigenvalues and eigenfunctions are in our disposal.

Inverse problems in waveguides were considered from the physical point of view in [9] and from the view point similar to ours in [5], [7].

2. Waveguide with cylindrical ends

In this paper, we provide a sketch of solution to the formulated inverse problem in the special case when the manifold(M,g)is a waveguide with cylindrical ends. This means that(M,g)can be represented as

M =M^c∪

N

[

α=1

(Ω^α×(0,∞))

!

, M^c∩(Ω^α×(0,∞))) = Ω^α×(0,1), Ω^α∩Ω^β =∅, forα6=β.

(2.1)

HereM^c is an open, relatively compact subset ofM and eachΩ^α×(0,∞) has a product structure

g|Ω^α×(0,∞)=

n−1

X

p,q=1

g_pq^α(x)dx^pdx^q+dy², whereX = (x, y)are the local coordinates on Ω^α×R+.

Note that the spectral and scattering theory for waveguides with cylin- drical ends much proceeds the general theory, see e.g. [10], [11], [17], [18], [22]. This is due to the fact that, in case of cylindrical ends, the scatter- ing problem for (M,g) can be considered as a perturbation in a compact domain of a direct sum of the Laplace operators,

∆0=

N

M

α=1

∆^α+∂_y² ,

where y ∈ R+ is the coordinate along each channel. In the case when

∂M =∅, the domain of definition of ∆₀ is

D(∆0) =⊕^N_α=1{uα∈H²(Ω^α×R+) : uα|Ω^α×{0}= 0},

and if∂M 6= 0, the∆₀is defined using the Dirichlet or Neumann boundary condition on∂Ω^α×(0,∞).

Let us return to the eigenfunctions of the continuous spectrum of ∆_M, ψ_m^α(X;k²), with m6d^α(k), k² ∈/ σ^s, which were introduced in Section 1.

Then, inΩ^β×(0,∞),

ψ^α_m(x^β, y;k²) =e^−iy

√

k²−λ^α_mφ^β_m(x^β)δα,β

+ X

l6d^β(k)

a^αβ_ml(k²)e^iy

pk²−λ^β_l φ^β_l(x^β)

+ X

l>d^β(k)

b^αβ_ml(k²)e^−y

pλ^β_l−k²φ^β_l(x^β).

(2.2)

Comparing the above formula with the asymptotic behavior (1.2) ofψ_m^α, we see that

a^αβ_ml(k²) =S^αβ_ml(k²), m6d^α(k), l6d^β(k).

Note that, in addition to the waves propagating towards infinity, which are represented by the first sum in the right side of (2.2), the wavefunction ψ_m^α contains, inΩ^β×(0,∞), infinitely many exponentially decaying terms, described by the second sum in the right side of (2.2). Therefore, it is possible to extend the notion of the scattering matrix from the physical oneS^αβml(k²), m6d^α(k), l6d^β(k), to the generall∈Z+. Namely, we first define the non-physical scattering matrixS_ml^αβ(k²)by

S^αβ_ml(k²) =a^αβ_ml(k²)form6d^α(k), l6d^β(k);

S^αβ_ml(k²) =b^αβ_ml(k²), form6d^α(k), l > d^β(k).

Moreover, it is also possible to treat the case ofm > d^α(k), that is,λ^α_m> k². To this end, we start with an exponentially growing, as y → ∞, incom- ing wave,exp (yp

λ^α_m−k²)φ^α_m(x^α) = exp (−iyp

k²−λ^α_m)φ^α_m(x^α). Starting from the above wave, we construct a non-physical wavefunctionψ^α_m(X;k²) by using the Green functionR_k2(X, X⁰)of(∆M +k²),

(2.3) ψ^α_m(X;k²)−χ(y) exp (yp

λ^α_m−k²)φ^α_m(x^α)

= Z ∞

0

Z

Ω^α

R_k2(X, X⁰) [∆⁰_M, χ(y⁰)] exp (y⁰p

λ^α_m−k²)φ^α_m(x^α)dy⁰dx^α, where X = (x, y), X⁰ = (x⁰, y⁰). Here χ(y) is a cut-off function, equal to 0 for y < 1/4 and 1 for y > 3/4 which vanishes in M \Ω^α×R+, [∆⁰_M, χ(y⁰)]stands for the commutator of the above operators, and∆⁰_M is the operator∆M in coordinates(x⁰, y⁰).

Next, denote the resolvent of−∆M byR_z= (−∆M−z)⁻¹. It is known, see e.g. [18] for the waveguides with cylindrical ends and [13] for the general compound waveguides, that, for anyΦ,Φe ∈C₀^∞(M),

R^Φ,e^Φ

k² :=m_Φ◦R_k2◦m

eΦ∈ L(L²(M)),

wherem_Φis the multiplication operator(m_Φu)(X) = Φ(X)u(X)andL(B) denotes bounded operators in a Banach space B. On the other hand, due toM being a compact domain perturbation ofSN

β=1 Ω^β×(0,∞)

, the op- erator[∆⁰_M, χ(y⁰)]has a compact support inΩ^α×[1/4,3/4]. Thus, equation (2.3) has sense, defining the non-physical wavefunctionψ_m^α(X;k²).

Using the limiting absorption principle for R_k2±iε, ε > 0, see e.g [13], where Rz is the resolvent for −∆M, we see that there are no propagating terms of the formexp (−iy

q

k²−λ^β_l), l6d^β(k)due to the integral in the right side of (2.3) . Summarizing, we see that, fory >3/4,

(2.4) ψ_m^α(x^β, y;k²) = exp (yp

λ^α_m−k²)φ^α_m(x^β)δαβ

+

∞

X

l=1

S_ml^αβ(k²) exp (iy q

k²−λ^β_l)φ^β_l(x^β),

thus defining the scattering matrix S_ml^αβ(k²) for m > d^α(k). In following, we denote by

S(k²) = [S_ml^αβ(k²)]_m,l∈Z+, α,β=(1,...,N)

this generalized, or the non-physical scattering matrix. Note that the phys- ical scattering matrixS(k²)is a finite dimensional sub-matrix ofS(k²)for eachk²∈R+\σ^s(M).

3. Analytic properties of the non-physical scattering matrix

To analyse the dependence ofS_ml^αβ(λ), λ∈C, we recall that the operator- valued functionR^Φ,Φe

λ is analytic inC\R+and has continuous limits,R^Φ,e^Φ

k²±i0

asλ→k²±i0(except fork²∈σ^s(M).)

Thus, equation (2.3) makes it possible to define the non-physical wave- functionψ^α_m(X;λ)for allλ∈C\R+. Considered as a function ofλ,ψ_m^α(·;λ) is analytic inC\R+ and, moreover, there are limits

ψ_m^{±, α}(X;k²) = lim

ε→0ψ^α_m(X;k²±iε).

Remark 3.1. — In our definition ofψ^α_m(X;k²), it corresponds actually toψ_m^{+, α}(X;k²)withψ^{−, α}_m (X;k²) =ψ^{+, α}_m (X;k²). Similarly,R_k2(X, X⁰)in (2.3) actually coincides withR_k2+i0(X, X⁰).

Therefore, wheny >0, Ψ^{s, αβ}_ml (y;λ) :=

ψ^α_m(·, y;λ)−exp (−iyp

λ−λm)φ^α_m(·)δαβ, φ^β_l(·)

L²(Ω^β)

satisfies, forλ∈C\R+, the ordinary differential equation d²

dy²Ψ^{s, αβ}_ml (y;λ) + (λ−λ^β_l)Ψ^{s, αβ}_ml (y;λ) = 0.

This and (2.4) imply that

Ψ^{s, αβ}_ml (y;λ) =S_ml^αβ(λ) exp (iy q

λ−λ^β_l),

whereS_ml^αβ(λ)is analytic inC\R+ and continuous, from above and below, up to R+, thus defining S_ml^αβ(k²±i0). Note that S^αβ_ml(k²) in section 2 is actuallyS^αβ_ml(k²+i0).

These considerations immediately imply the following lemma:

Lemma 3.2. — Let S(k²) be given for all k² ∈ O, where O is an un- bounded open subset ofR+,O ∩σ^s(M) =∅. Then, these data determine the non-physical scattering matrix S_ml^αβ(λ), m, l ∈ Z+, α, β ∈ {1, . . . , N} for allλ∈C\R+ and alsoS_ml^αβ(k²±i0)for allk²∈/ σ^s(M).

ProofObserve that for anym, landα, β, there is an open intervalI⊂O such that, fork²∈I, we haveλ^α_m, λ^β_l < k². Moreover, for suchk²,

S^α,βml(k²) =S_ml^αβ(k²+i0).

By the analyticity properties ofS^αβml(λ), described above, this determines uniquely this matrix coefficient inC\R+ and also S_ml^αβ(k²±i0) in R+\ σ^s(M). QED

Note that having only a partial physical scattering matrix S^ph forα, β lying in a subset K of (1, . . . , N), we can find the partial non-physical matrixS_ml^αβ(λ)for theseα, β.

4. From scattering data to the Dirichlet-to-Neumann map In this and the next sections we consider the case when the physical scattering matrixSis known only in one channel of scatteringΩ^α×(0,∞), corresponding to, say, α = 1, i.e. we are given S^αα_m,l(k²), m, l 6 d^α(k)

with α= 1. In this connection, in the future we skip indexes α, β writing justΩ = Ω¹, Sml(λ) =S_ml¹¹(λ), ψm(X, k²) =ψ¹_m(X, k²), φm(x) =φ¹_m(x), etc. We denote byMfthe domain

Mf=M \(Ω×(1,∞)).

Observe that, in general, (M ,f g|

Me) is itself a compound waveguide with Ω× {1} ⊂∂Mf.

Consider the Dirichlet-to-Neumann operatorΛeλ, λ∈C\R+ associated withMf,

(4.1) Λeλ:H^1/2(Ω)→H^−1/2(Ω); Λeλ(f) :=∂yu^f_λ|_(Ω×{1}), whereu^f_λ(X)is the solution to the boundary-value problem,

(e∆ +λ)u^f = 0 in Mf u^f|_Ω×{1}=f,

where∆e is the Laplace operator inMf.

Then, the partial non-physical scattering matrixSdetermines the action ofΛeλ, forλ∈C\R+, on all functionsf, such that

(4.2) f ∈cl_H1/2(Ω){Span(ψ_m(x,1;λ), m= 1,2, . . .)}.

Indeed, forf(x) =ψ_m(x,1;λ), we have,

u^f_λ(X) =ψm(X;λ), X ∈M .f Thus,

∂yu^f_λ(x)|_(Ω×{1})=−ip

λ−λmexp (−ip

λ−λm)φm(x) +

∞

X

l=1

ip

λ−λlSml(λ) exp(ip

λ−λl)φl(x).

(4.3)

Here, due to the orthogonality,(φm, φl)_H1(Ω)= 0, form6=l, and regularity ψm ∈ H_loc² (M), the sum in (4.3) converges in H¹(Ω). Note that we also know that

ψm|_Ω×{1}= exp (−ip

λ−λ_m)φ_m(x) +

∞

X

l=1

S_ml(λ) exp(ip

λ−λ_l)φ_l(x).

(4.4)

Then the desired result thatΛe_λon the set (4.2) can be determined using the scattering matrix S_ml(λ)follows from the continuity of Λe_λ from H^1/2(Ω) toH^−1/2(Ω).

Therefore, if we can show that

(4.5) cl_H1/2(Ω{Span(ψm(x,1;λ), m= 1,2, . . .)}=H^1/2(Ω),

then the partial non-physical scattering matrixS(λ)determines the Diri- chlet-to-Neumann mapΛe_λ. In turn, by duality, property (4.5) is equivalent to the following result

Lemma 4.1. — Leth∈H^−1/2(Ω). Assume that, for someλ∈C\R+, hh, ψm(λ)|_Ω×{1}i= 0, m= 1,2, . . .

Thenh= 0.

(In the above formulah·,·istands for the sesquilinearH^−1/2(Ω)×H^1/2(Ω) duality.)

Proof. — Let us return to representation (2.3) which we rewrite slightly differently as

ψ_m(X;λ) = 2iχ(y) sin (yp

λ−λ_m)φ_m(x) + 2i

Z ∞

0

Z

Ω

R_λ(X, X⁰) [∆⁰_M, χ(y⁰)] sin (y⁰p

λ−λ_m)φ_m(x⁰)dy⁰dx⁰. (4.6)

Here we take into the account that the non-physical wavefunction in the

“unperturbed waveguide”, M⁰ = Ω×(0,∞) with Dirichlet condition at Ω× {0}, which corresponds to {λm, φm}, is sin (y√

λ−λm)φm(x). Next, let X = (x, y), X⁰ = (x⁰, y⁰), X⁰⁰ = (x⁰⁰, y⁰⁰) be points of M⁰. Observe, that when y⁰⁰ > 1, the Green function Rλ(X, X⁰⁰) does itself satisfy a representation similar to (4.6),

R_λ(X, X⁰⁰) =χ(y)R⁰_λ(X, X⁰⁰) +

Z ∞

0

Z

Ω

R_λ(X, X⁰) [∆⁰_M, χ(y⁰)]R⁰_λ(X⁰, X⁰⁰)dy⁰dx⁰, (4.7)

where R⁰_λ(X, X⁰⁰) is the Green function in Ω×(0,∞) with the Dirichlet condition atΩ× {0}. We have, forX = (x, y),X⁰= (x⁰, y⁰),y < y⁰, that (4.8)

R⁰_λ(X, X⁰) =

∞

X

m=1

sin (yp

λ−λm) exp (iy⁰p

λ−λm)φm(x)φm(x⁰).

Therefore, substituting (4.8) into representation (4.7), we get R_λ(X, X⁰⁰)

=χ(y)

∞

X

m=1

sin (yp

λ−λm) exp (iy⁰⁰p

λ−λm)φm(x)φm(x⁰⁰)

+ Z ∞

0

Z

Ω

Rλ(X, X⁰) [∆⁰_M, χ(y⁰)]

×

∞

X

m=1

sin (y⁰p

λ−λ_m) exp (iy⁰⁰p

λ−λ_m)φ_m(x⁰)φ_m(x⁰⁰)dy⁰dx⁰. Comparing the above representation with (4.6), we see that, for X = (x, y)∈M , Xf ⁰⁰= (x⁰⁰, y⁰⁰), y⁰⁰>1,

(4.9) Rλ(X, X⁰⁰) =

∞

X

m=1

ψm(X;λ) exp (iy⁰⁰p

λ−λm)φm(x⁰⁰).

Let nowh∈H^−1/2(Ω)be orthogonal to allψ_m(·;λ)|_(Ω×{1}), i.e.

(4.10) hh, ψ_m|_Ω×{1}i:=

Z

Ω

h(x)ψ_m(x,1;λ)dx= 0, m= 1,2, . . . Consider the equation

(∆_M +λ)u=h(x)δ(y−1)∈H⁻¹(M).

Forλ∈C\R+, it has the solution of the form of a single-layer potential, (4.11) u^h(X) =

Z

Ω

R_λ(X; (x⁰,1))h(x⁰)dx⁰, u^h∈H¹(M).

Let nowX = (x, y), y >1. Observe that

R_λ(X⁰, X) =Rλ(X, X⁰).

Therefore, condition (4.10) imply that, fory >1, u^h(x, y) =

∞

X

m=1

hh, ψ_m|_Ω×{1}i

exp (−iyp

λ−λ_m)φ_m(x) = 0,

where we choose the branch of √

z to be equal to −i|z|^1/2 for z ∈ R−. As u^h ∈ H¹(M) this means, in particular, that u^h|

Me is solution to the homogeneous Dirichlet problem. Asλ /∈R+, this implies thatu^h= 0inMf and, therefore,h= 0. QED

Summarizing the above, we formulate the following result.

Lemma 4.2. — Let(M,g)be a waveguide with cylindrical ends. Then, for anyαthe partial physical scattering matricesS^ααml(k²), m, l6d^α(k), k∈ R+determines uniquely, for anyλ∈C\R+, the Dirichlet-to-Neumann map Λeλ of the Laplace operator,∆e in Mf.

5. Main result

We are now in the position to formulate the main result of the pa- per. To this end, consider two waveguides, each with cylindrical ends, (M1,g1),(M2,g2). Assume that for some ends, say, those numbered by α = 1, the boundaries of these ends at infinity, (Ω^α₁, g₁^α),(Ω^α₂, g^α₂) with α= 1, are isometric. Next we omit αand write just Ω¹₁ = Ω1, Ω¹₂ = Ω2, etc. Identifying them, we write

(5.1) (Ω1, g1) = (Ω2, g2) = (Ω, g).

Consider the Laplace operators ∆M_i in(Mi,gi), i= 1,2, and the corre- sponding partial physical scattering matrices in channelsΩ₁×(0,∞),Ω₂× (0,∞), namelyS¹¹_(i)(k²) = [S¹¹_(i),ml(k²)]_m,l6d1(k),.

Theorem 5.1. — Let(M1,g1),(M2,g2) satisfy the above conditions.

Assume that, for anyk²∈O,whereOis an unbounded open set inR+such thatO∩σ^s(Mi) =∅, the physical scattering matrices corresponding to the endΩ×R+, S¹¹₍₁₎(k²)and S¹¹₍₂₎(k²), k² ∈O, coincide. Then the manifolds (fM1,g1),(Mf2,g2) are isometric. In particular, the number of scattering channels is the same,N₁ =N₂ (= N) and, after a proper identification, (Ω^α₁, g₁^α)are isometric to (Ω^α₂, g₂^α), α= 1, . . . , N.

Proof. — By Lemma 3.2, the conditions of the theorem imply that S_(1),ml(k²) =S_(2),ml(k²)

for allm, l∈Z+, k²∈O. Denoting byψm⁽ⁱ⁾(X, λ)the wavefunctions, both physical and non-physical, ofMi, i= 1,2,and using equations (4.3), (4.4), we see that

ψ_m⁽¹⁾(x,1;k²) =ψ_m⁽²⁾(x,1;k²),

∂_yψ_m⁽¹⁾(x,1;k²) =∂_yψ_m⁽²⁾(x,1;k²);

x∈Ω, k²∈O.

Due to the analyticity, with respect toλ, ofψm⁽ⁱ⁾(·, λ), see the beginning of section 3, we see that

ψ⁽¹⁾_m(x,1;λ) =ψ⁽²⁾_m(x,1;λ),

∂_yψ⁽¹⁾_m(x,1;λ) =∂_yψ⁽²⁾_m(x,1;λ);

x∈Ω, λ /∈R+.

It then follows from Lemma 4.2 that the Dirichlet-to-Neumann maps forMf1

andMf₂coincide,

(5.2) Λe⁽¹⁾_λ =Λe⁽²⁾_λ . Consider now the wave equations inMf_i×R,

(5.3) ∂²_tuⁱ_F−∆eiuⁱ_F = 0, uⁱ_F|t<0= 0, uⁱ_F|_Ω×{1}=F.

It follows from (5.2), cf. [15], that

(5.4) Λe⁽¹⁾_h (F) =Λe⁽²⁾_h (F), F ∈C₀^∞(Ω×R⁺), whereΛe⁽ⁱ⁾_h is the hyperbolic Dirichlet-to-Neumann map,

Λe⁽ⁱ⁾uⁱ_h(F) =∂yuⁱ_F|_{(Ω×{1})×R+}. This implies, it turn, that there exists an isometry

Φ: (Mf1,g1)→(Mf2,g2), Φ|_(Ω×{1})=Id|_(Ω×{1}), see e.g. [16].

AsMfi∩(Ω×R+) = (Ω×(0,1)),the second equation in the above formula yields also that(M1,g1)and(M2,g2)are isometric. QED Remark 5.2. — Instead of given isometric Ω₍₁₎,Ω₍₂₎, we can assume that the spectra of∆_M1 and∆_M2 coincide as well as the sets

Φⁱ:={(φⁱ₁|_Ω(i), φⁱ₂|_Ω(i), . . .)} i= 1,2, cf. [1].

Remark 5.3. — The general theory [2], [3], [19] describes spectral and scattering properties of compound waveguides with asymptotically cylindri- cal ends. Also, constructions in [16] remain valid for the manifolds(M ,f eg) with asymptotically cylindrical ends. Therefore, the constructions of the paper, in particular Theorem 5.1, remain valid even if we do not assume that the ends ofM1and M2, except forΩ×(0,∞), are cylindrical.

Remark 5.4. — The idea to consider, in studying inverse scattering, not only the "physical" wavefunctions, but also "non-physical", exponentially growing ones, goes back to [8], which used it to study inverse quantum scattering with data given by the scattering matrix at all energies. Later, this type of exponentially growing solutions proved most effective in the study of inverse problems with fixed-frequency data, see [21], [20], [14] for the pioneering works. In this paper we return, for the waveguide inverse problem, to the original idea of Faddeev and combine it with the technique of the BC-method, see e.g. [15].

Acknowledgements. — M.L. has been partially supported by Tekes pro- ject MASIT03, and the Academy of Finland Center of Excellence pro- gramme 213476.

BIBLIOGRAPHY

[1] Berard P., Besson G., Gallot S.Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal,4(1994), 373-398.

[2] Christiansen T., Scattering theory for manifolds with asymptotically cylindrical ends, J. Funct. Anal.131(1995), 499-530.

[3] Christiansen T. and Zworski R.,Spectral asymptotics for m anifolds with cylindrical ends, Ann. Inst. Fourier (Grenoble),45(1995), 251-263.

[4] Colin de Verdière, Y.,Une nouvelle démonstration du prolonement méromorphe des série d’ Eisenstein,C. R. Acad. Sc. Paris, t.293(1981), 361-363.

[5] Dediu S., McLaughlin J.Recovering inhomogeneities in a wave guide using eigen- system decomposition,Inverse Problems32(2006), 1226-1246.

[6] Eidus D.,Completeness properties of scattering problem solutions.Comm. PDE7 (1982), 55–75.

[7] Eskin G., Ralston J., Yamamoto M., Inverse scattering for gratings and wave guides.Preprint: arXiv:0704.2274v1

[8] Faddeev L.D.,The inverse problem in the quantum theory of scattering. II.(Rus- sian) Current problems in mathematics,3(Russian), (1974), 93–180.

[9] Gilbert R.P., Mawata C., Xu Y.,Determination of a distributed inhomogeneity in a two-layered waveguide from scatteed sound, in: Direct and Inverse Problems of Mathematical Physics, ed. Gilbert R. et al, Kluwer, 2000.

[10] Goldstein C.,Eigenfunction expansions associated with the Laplacian for certain domains with infinte boundaries, Trans. A. M. S.135(1969), 1-50.

[11] Goldstein C.,Meromorphic continuation of the S-matrix for the operator−∆ac- tying in a cylinder, Proc. A. M. S.42(1974), 555-562.

[12] Isakov V. and Nachman A.,Global uniqueness for a two-dimensional semilinear elliptic inverse problem,Trans. Amer. Math. Soc.347(1995), 3375–3390

[13] Isozaki H., Kurylev Y. and Lassas M., Spectral and scattering properties for the compound waveguides with asymptotially cylindirical ends, in preparation.

[14] Khenkin G.M., Novikov R.,The∂-equation in the multidimensional inverse scat- tering problem.(Russian), Usp. Mat. Nauk42(1987), 93–152

[15] Kachalov A., Kurylev Ya., Lassas M., Inverse Boundary Spectral Problems, Chap- man Hall / CRC123, 2001.

[16] Kachalov A., Kurylev Y., Lassas M., Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds. IMA volumes in Mathematics and Applications (Springer Verlag) Geometric Methods in Inverse Problems and PDE Control, Ed. C. Croke, I. Lasiecka, G. Uhlmann, M. Vogelius, 2004, pp. 183-214.

[17] Lyford W.C.,Spectral analysis of the Laplacian in domains with cylinders,Math.

Ann,218(1975), 229-251.

[18] Lyford W.C.,Asymptotic energy propagation and scattering of waves in waveguides with cylinders,Math. Ann,219(1976), 193-212.

[19] Melrose R. B.,Geometric Scattering Theory. Cambridge University Press, Cam- bridge (1995), 116 pp.

[20] Nachman A., Sylvester J., Uhlmann G.,Ann-dimensional Borg-Levinson theorem.

Comm. Math. Phys.115(1988), 595–605.

[21] Sylvester J., Uhlmann G., A global uniqueness theorem for an inverse boundary value problem.Ann. Math.,125(1987), 153–169.

[22] Wilcox C., Spectral and asymptotic analysis of acoustic wave propagation, in:

Boundary value problems for linear evolution: partial differential equations, NATO Advanced Study Inst. Ser., Ser. C: Math. and Phys. Sci., 29, Reidel, Dordrecht, (1977), 386-473.

HiroshiIsozaki University of Tsukuba Institute of Mathematics Tsukuba, 305-8571 (Japan) YaroslavKurylev

University College of London Department of Mathematics United Kingdom

MattiLassas

Helsinki University of Technology Department of Mathematics Finland