INVERSE PROBLEMS FOR THE PAULI HAMILTONIAN IN TWO DIMENSIONS

HYEONBAE KANG AND GUNTHER UHLMANN

Abstract. We prove in two dimensions that the set of Cauchy data for the Pauli Hamiltonian measured on the boundary of a bounded open subset with smooth enough boundary determines uniquely the magnetic field and the electrical potential provided that the electrical potential is small in an appropriate topology. This result has the immediate consequence, in the case that the magnetic potential and electrical potential have compact support, that we can determine uniquely the magnetic field and the electrical potential by measuring the scattering amplitude at a fixed energy provided that the electrical potential is small in an appropriate topology.

Mathematics subject classification (MSC2000): 35R30 Keywords: Pauli Hamiltonian, Cauchy data, Dirichlet-to-Neumann map, inverse scattering

1. Introduction

The Pauli Hamiltonian describes particles in a magnetic field with spin. In two dimensions it is a direct sum of the pair of operators

(1.1)  H_{A⃗,q}u := ∑_{j=1}^{2} (\frac{1}{i} \frac{∂}{∂x_{j}} − A_{j})^{2} u ± Bu − qu

where A⃗ denotes the magnetic potential, B = rotA⃗ is the magnetic field, and q is the electrical potential (see for instance Chapter 6 of [4]). Thus both direct and inverse problems for the Pauli Hamiltonian consider separately both signs in B in (1.1). For simplicity we will choose the plus sign. All of the results below are also valid for the minus sign in B with minor changes in the arguments.

We first describe the inverse boundary problem we consider. Let Ω ⊂ ℝ^{2} be a bounded domain with smooth boundary. We consider the Pauli Hamiltonian given by a real-valued vector field A⃗ = (A_{1}, A_{2}) ∈ W^{1,p}(Ω) and an electric potential q ∈ L^{p}(Ω), p > 2,

(1.2)  H_{A⃗,q}u := ∑_{j=1}^{2} (\frac{1}{i} \frac{∂}{∂x_{j}} − A_{j})^{2} u + Bu − qu = 0 in Ω

Both authors are grateful to the Mathematical Sciences Research Institute (MSRI) for partial support and for providing a very stimulating environment during the inverse problems program in fall 2001. HK is partly supported by KOSEF 98-0701-03-5 and 2000-1-10300-001-1. GU is partly supported by NSF grant DMS-0070488 and a John Simon Guggenheim fellowship.

where B = rotA⃗. Fix α = \frac{p−2}{p} throughout this paper. The set of Cauchy data of the solutions of (1.2) is given by

(1.3)  C_{A⃗,q} := \{(f, g) ∈ C^{1,α}(∂Ω) × C^{α}(∂Ω) : there exists u ∈ C^{1,α}(\overline{Ω})
such that H_{A⃗,q}u = 0, u|_{∂Ω} = f, ((∇ − iA⃗)u)|_{∂Ω} · ν = g}.

Here ν denotes the unit normal to ∂Ω. In the case that 0 is not a Dirichlet eigenvalue for H_{A⃗,q}, C_{A⃗,q} is the graph of the Dirichlet-to-Neumann (DN) map

(1.4)  Λ_{A⃗,q} : C^{1,α}(∂Ω) → C^{α}(∂Ω).

The inverse boundary value problem we consider in this paper is whether we can determine A⃗ and q from C_{A⃗,q}.

It was observed in [9, 10] that there is a gauge invariance in the problem. That is, if φ ∈ C^{1}(\overline{Ω}) with φ|_{∂Ω} = 1, ∇φ|_{∂Ω} = 0, then

C_{A⃗+∇φ,q} = C_{A⃗,q}.

Therefore we can recover at best the magnetic field, rotA⃗, and q from the DN map. In this paper we prove the following semiglobal identifiability result:

Theorem 1.1. Let A⃗_{j} ∈ W_{0}^{1,p}(Ω), j = 1, 2, rotA⃗_{1} ∈ W_{0}^{1,p}(Ω), q_{1} ∈ W^{1,p}(Ω), q_{2} ∈ L^{p}(Ω), p > 2. For each M > 0, there exists ϵ(M, Ω, p) > 0 such that if ∥A⃗_{1}∥_{L^{p}(Ω)} ≤ M and ∥q_{1}∥_{W^{1,p}(Ω)} ≤ ϵ and

(1.5)  C_{A⃗_{1},q_{1}} = C_{A⃗_{2},q_{2}},

we conclude

(1.6)  rotA⃗_{1} = rotA⃗_{2} and q_{1} = q_{2} in Ω.

W_{0}^{1,p}(Ω) denotes the space of W^{1,p}(Ω)-functions whose boundary traces are zero. Observe that no smallness condition is assumed on the electric potential q_{2}. We also

remark that the only place where we need that the magnetic potential is compactly supported in Ω is in the proof of Lemma 4.1. If we assume further regularity in the magnetic and electrical potentials then Theorem D of [NSU] allows to extend the magnetic and electrical potentials to ℝ^{2} with compact support. More precisely we have:

Theorem 1.2. Let A⃗_{j}, q_{j} ∈ C^{∞}(\overline{Ω}), j = 1, 2, and p > 2. There exists ϵ(Ω, p) > 0 such that if ∥q_{1}∥_{W^{1,p}(Ω)} ≤ ϵ and

(1.7)  C_{A⃗_{1},q_{1}} = C_{A⃗_{2},q_{2}},

we conclude

(1.8)  rotA⃗_{1} = rotA⃗_{2} and q_{1} = q_{2} in Ω.

In [10] Sun proved in two dimensions for the Schrödinger equation in a magnetic field that if ∥rotA⃗_{j}∥_{W^{1,∞}(Ω)} (j = 1, 2) is small enough and q_{j} (j = 1, 2) are in an open and dense set in an appropriate topology, then we can determine uniquely rotA⃗_{j} and q_{j} from the DN map associated to the magnetic potentials and electrical potentials.

We remark that in dimensions n ≥ 3 a global identifiability result of the magnetic field and electrical potential was proven in [8] for the Schrödinger equation in a magnetic field assuming some smoothness conditions on the coefficients. We also note that even if we assume that A⃗_{1} = 0 in Theorem 1.1, it is unknown whether we can recover q from C_{q} := C_{0,q}, the set of Cauchy data for the standard Schrödinger equation, although several partial results have been proven ([3], [5], [6], [11], [12], [16]).

A particular case of Theorem 1.1 is when the electrical potential in (1.1) is zero. Thus we obtain the following global uniqueness result:

Corollary 1.3. Let A⃗_{j} ∈ W_{0}^{1,p}(Ω), j = 1, 2, rotA⃗_{1} ∈ W_{0}^{1,p}(Ω), q_{1} = 0, q_{2} ∈ L^{p}(Ω), p > 2. For each M > 0, there exists ϵ(M, Ω, p) > 0 such that if ∥A⃗_{1}∥_{L^{p}(Ω)} ≤ M

(1.9)  C_{A⃗_{1},0} = C_{A⃗_{2},q_{2}},

we conclude

(1.10)  rot A_{1} = rot A_{2} and q_{1} = q_{2} = 0 in Ω.

As a consequence of Theorem 1.2, a similar result holds with A⃗_{j} ∈ C^{∞}(\overline{Ω}), j = 1, 2, without the assumption that the magnetic potentials are compactly supported in Ω.

In the case that the magnetic potentials are zero, we conclude the following semiglobal identifiability result:

Corollary 1.4. Let q_{1} ∈ W^{1,p}(Ω), q_{2} ∈ L^{p}(Ω), p > 2. There exists ϵ(Ω, p) > 0 such that if ∥q_{1}∥_{W^{1,p}(Ω)} ≤ ϵ and C_{q_{1}} = C_{q_{2}}, then q_{1} = q_{2}.

Corollary 1.4 was known previously if q_{1} and q_{2} are both a-priori close to constant ([16]). As far as we know, even the case q_{1} = 0 was previously unknown.

It is well known (see for instance chapter 12 of [17], and [7]) that Theorem 1.1 implies a similar result for the inverse scattering problem at fixed energy if we assume that the magnetic potential and electrical potential have compact support.

The scattering amplitude for the Pauli Hamiltonian (1.1) with A⃗ ∈ W^{1,p}(ℝ^{2}), q ∈ L^{p}(ℝ^{2}), p > 2, A⃗, q with compact support, is defined in terms of the outgoing eigenfunctions ψ_{+}(λ, x, ω) where λ ∈ ℝ − 0, x ∈ ℝ^{2}, ω ∈ S^{1}. Namely ψ_{+} satisfies

ψ_{+}(λ, x, ω) = e^{iλx·ω} + \frac{a_{A⃗,q}(λ, θ, ω)}{|x|^{\frac{1}{2}}}e^{iλ|x|} + O(|x|^{− \frac{3}{2}} ),

where θ = \frac{x}{|x|}. The scattering amplitude, a_{A⃗,q}(λ, θ, ω), measures the effect of the magnetic and electrical potential on a plane wave with frequency λ and direction ω of the form e^{iλx·ω}.

The inverse scattering problem at a fixed energy is to determine A⃗, q from the scattering amplitude a_{A⃗,q}(λ_{0}, θ, ω) with fixed λ_{0}. It is easy to see, as in the case of the Cauchy data previously discussed, that we can recover at best the magnetic field and the electrical potential. An immediate consequence of Theorem 1.1 is the following.

Theorem 1.5. Let A⃗_{j} and q_{j} have compact supports. Let A⃗_{j} ∈ W^{1,p}(ℝ^{2}), j = 1, 2, rotA⃗_{1} ∈ W^{1,p}(ℝ^{2}), q_{1} ∈ W^{1,p}(ℝ^{2}), q_{2} ∈ L^{p}(ℝ^{2}), p > 2. For each M > 0, there exists ϵ(M, Ω, p) > 0 such that if ∥A⃗_{1}∥_{L^{p}(Ω)} ≤ M and ∥q_{1}∥_{W^{1,p}(Ω)} ≤ ϵ and

(1.11)  a_{A⃗_{1},q_{1}} = a_{A⃗_{2},q_{2}} for a fixed λ,

then

(1.12)  rotA⃗_{1} = rotA⃗_{2} and q_{1} = q_{2} in ℝ^{2}.

One can also state colloraries similar to Corollary 1.3 and Corollary 1.4.

The method of proof of Theorem 1.1 is by reducing the problem to a similar one for a second order equation which can be factored in terms of \overline{∂} and ∂. Recall

\overline{∂} = \frac{1}{2}(∂_{x_{1}} + i∂_{x_{2}}), ∂ = \frac{1}{2}(∂_{x_{1}} − i∂_{x_{2}}).

We rewrite (1.2) in the form

(1.13)  (\overline{∂} + ā)(∂ − a)u − q̃u = 0 in Ω

where

(1.14)  a := \frac{1}{2}(A_{2} + iA_{1}), q̃ = \frac{1}{4}q.

We define the set of Cauchy data associated to (1.13) by

(1.15)  C_{a,q̃} := {(f, g) ∈ C^{1,α}(∂Ω) × C^{α}(∂Ω) : u|_{∂Ω} = f,
((∂ − a)u)|_{∂Ω} = g, u ∈ C^{1,α}(\overline{Ω}) a solution of (1.13) \}.

Theorem 1.1 is then a consequence of

Theorem 1.6. Let a_{j} ∈ W^{1,p}(Ω), q̃_{1} ∈ W^{1,p}(Ω), q̃_{2} ∈ L^{p}(Ω), p > 2, j = 1, 2. For each M > 0, there exists ϵ(M, Ω, p) > 0 such that if ∥a_{1}∥_{L^{p}(Ω)} ≤ M and ∥q̃_{1}∥_{W^{1,p}(Ω)} ≤ ϵ and

(1.16)  C_{a_{1},q̃_{1}} = C_{a_{2},q̃_{2}},

then

(1.17)  q̃_{1} = q̃_{2} and \overline{∂}^{−1}ā_{1} + ∂^{−1}a_{1} = \overline{∂}^{−1}ā_{2} + ∂^{−1}a_{2} in Ω.

Of course (1.17) implies that

(1.18)  rota_{1} = \frac{1}{2}(\overline{∂}ā_{1} + ∂a_{1}) = rota_{2} = \frac{1}{2}(\overline{∂}ā_{2} + ∂a_{2}) in Ω.

The method of proof of Theorem 1.6 reduces (1.13) to a first order system and follows the lines of [3] to construct complex geometrical optics solutions for (1.13) for all complex frequencies, and uses the scattering transform of Beals and Coifman [2] (see also [13, 14, 15]). An important difference with [3] is that we work directly on the bounded domain Ω rather than the whole space as in [3]. This simplifies several of the arguments.

In section 2 we construct the complex geometric optics solutions for the first order system and consider the corresponding scattering transform. In section 3 we derive the scattering relation of the solutions. Finally in section 4 we prove Theorem 1.6.

2. Complex geometrical optics solutions for the \overline{∂}-system

In this section we reduce the second order equation

(2.1)  (\overline{∂} + ā)(∂ − a)u − qu = 0 in Ω

into a system of first order \overline{∂} type system and construct geometrical optics solutions of the system.

Define the solution operator \overline{∂}^{−1} of the \overline{∂}-equation by

(2.2)  \overline{∂}^{−1}f(z) := −\frac{1}{π} ∫_{Ω} \frac{f(ζ)}{ζ − z} dμ(ζ),  z ∈ Ω

where dμ is the Lebesgue measure on ℝ^{2}. We note that \overline{∂}^{−1} in this paper is defined as an integral operator on Ω, not on the whole ℝ^{2}. Then it is well-known that

(2.3)  f(z) = \overline{∂}^{−1}(\overline{∂}f)(z) + C(f|_{∂Ω})(z), z ∈ Ω

where C(f|_{∂Ω}) is the Cauchy transform of f|_{∂Ω}, namely,

C(f|_{∂Ω})(z) = \frac{1}{2πi} ∫_{∂Ω} \frac{f(ζ)}{ζ − z} dζ, z ∉ ∂Ω.

We then define ∂^{−1} and \overline{C} to be the conjugates of \overline{∂}^{−1} and C, respectively, i.e.,

∂^{−1}(f) := \overline{\overline{∂}^{−1}(f̄)}, and \overline{C}(f) := \overline{C(f̄)}.

Then the following formula immediately follows from (2.3):

(2.4)  f(z) = ∂^{−1}(∂f)(z) + \overline{C}(f|_{∂Ω})(z), z ∈ Ω.

We also define

(2.5)  C := ( \begin{matrix}C & 0 \\ 0 & \overline{C} \end{matrix} ) .

Let u be a solution of (2.1) and set w := (∂ − a)u. Then the equation (2.1) takes the form

(2.6)  [( \begin{matrix}\overline{∂} + ā & 0 \\ 0 & ∂ − a \end{matrix} ) − ( \begin{matrix}0 & q \\ 1 & 0 \end{matrix} )] ( \begin{matrix}w \\ u \end{matrix} ) = 0.

By conjugating the equation (2.6), we have (2.7)

( \begin{matrix}e^{−\overline{∂}^{−1}ā} & 0 \\ 0 & e^{∂^{−1}a} \end{matrix} ) [( \begin{matrix}\overline{∂} & 0 \\ 0 & ∂ \end{matrix} ) − ( \begin{matrix}0 & e^{T a}q \\ e^{−T a} & 0 \end{matrix} )] ( \begin{matrix}e^{\overline{∂}^{−1}ā} & 0 \\ 0 & e^{−∂^{−1}a} \end{matrix} ) ( \begin{matrix}w \\ u \end{matrix} ) = 0

where

T a(z) := \overline{∂}^{−1}ā + ∂^{−1}a.

Set

D = ( \begin{matrix}\overline{∂} & 0 \\ 0 & ∂ \end{matrix} )

and

Q = ( \begin{matrix}0 & e^{T a}q \\ e^{−T a} & 0 \end{matrix} ) .

We are seeking special solutions of the system

(2.8)  (D − Q)ψ = 0 in Ω

in the form

(2.9)  ψ = ( \begin{matrix}e^{−\overline{∂}^{−1}ā} & 0 \\ 0 & e^{∂^{−1}a} \end{matrix} ) m(z, k) ( \begin{matrix}e^{izk} & 0 \\ 0 & e^{−iz̄k} \end{matrix} )

where m(z, k) is a 2 × 2 matrix valued function in Ω.

We need a few more definitions (see [3]): For z and k in ℂ, let

ẽ_{k}(z) = exp(i(zk̄ + z̄k)) and e_{k}(z) = exp(i(zk + z̄k̄)),

and

Λ(z, k) = Λ_{k}(z)

= ( \begin{matrix}ẽ_{k}(z) & 0 \\ 0 & e_{−k}(z) \end{matrix} ) .

We then define

E_{k}A = E_{k} ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) = ( \begin{matrix}a_{11} & ẽ_{−k}a_{12} \\ e_{k}a_{21} & a_{22} \end{matrix} ) .

Then, m defined in (2.9) satisfies

(2.10)  D_{k}m − Qm = 0 in Ω

where D_{k} is the operator

D_{k}A = E_{k}^{−1}DE_{k}A.

Let

D^{−1} := ( \begin{matrix}\overline{∂}^{−1} & 0 \\ 0 & ∂^{−1} \end{matrix} ) .

We have from (2.10) that

DE_{k}m − E_{k}Qm = 0 in Ω.

Applying D^{−1} to the last equation and using (2.3) and (2.4), we get

E_{k}m(z) − D^{−1}E_{k}Qm(z) = C(E_{k}m|_{∂Ω})(z), z ∈ Ω.

Thus we conclude

(2.11)  (I − D_{k}^{−1}Q)m(z) = E_{k}^{−1}C(E_{k}m|_{∂Ω})(z), z ∈ Ω,

where I is the 2 × 2 identity matrix and

D_{k}^{−1} := E_{k}^{−1}D^{−1}E_{k}.

In order to investigate the invertability of the operator I − D_{k}^{−1}Q, we consider it’s null space. We denote by H(Ω) the class of holomorphic functions in Ω.

Lemma 2.1. Suppose that m ∈ L^{∞}(Ω) and satisfies (I−D_{k}^{−1}Q)m = 0 in Ω. Define ψ by (2.9). Then ψ_{21}, ψ_{22} ∈ C^{1,α}(\overline{Ω}) are solutions of the equation (\overline{∂}+ā)(∂−a)u−qu = 0 in Ω and there are f_{ij} ∈ H(ℂ \ \overline{Ω}) ∩ (ℂ \ Ω) with f_{ij}(z) = O(|z|^{−1}) as |z| → ∞, i, j = 1, 2, such that

(2.12)  \begin{cases}ψ_{21}|_{∂Ω} & = e^{∂^{−1}a}e^{−iz̄k̄}f̄_{21}, \\ (∂ − a)ψ_{21}|_{∂Ω} & = e^{−\overline{∂}^{−1}ā}e^{izk}f_{11}, \end{cases}

and

(2.13)  \begin{cases}ψ_{22}|_{∂Ω} & = e^{∂^{−1}a}e^{−iz̄k}f̄_{22}, \\ (∂ − a)ψ_{22}|_{∂Ω} & = e^{−\overline{∂}^{−1}ā}e^{izk̄}f_{12}. \end{cases}

The converse is also true: If ψ_{21}, ψ_{22} ∈ C^{1,α}(\overline{Ω}) are solutions of (2.1) and satisfy (2.12) and (2.13), then define ψ by ψ_{11} := (∂ − a)ψ_{21} and ψ_{12} := (∂ − a)ψ_{22}, and m by (2.9). Then we have (I − D_{k}^{−1}Q)m = 0 in Ω.

Proof. Suppose that m ∈ L^{∞}(Ω) and

(I − D_{k}^{−1}Q)m = 0 in Ω.

Then, m ∈ C^{1,α}(\overline{Ω}) and satisfies

(D_{k} − Q)m = 0 in Ω,

and, by (2.11),

C(E_{k}m|_{∂Ω})(z) = 0, z ∈ Ω.

Thus we obtain (2.14) C(m_{11}|_{∂Ω})(z) = \overline{C}(e_{k}m_{21}|_{∂Ω})(z) = C(ẽ_{−k}m_{12}|_{∂Ω})(z) = \overline{C}(m_{22}|_{∂Ω})(z) = 0, z ∈ Ω.

(2.14) is equivalent to the fact that there are f_{ij} ∈ H(ℂ \ \overline{Ω}) ∩ (ℂ \ Ω) with f_{ij}(z) = O(|z|^{−1}) as |z| → ∞, i, j = 1, 2, such that

m_{11}|_{∂Ω} = f_{11},
e_{k}m_{21}|_{∂Ω} = \overline{f_{21}},
ẽ_{−k}m_{12}|_{∂Ω} = f_{12},
m_{22}|_{∂Ω} = \overline{f_{22}}.

This can be proven using Plemelj’s jump formula for the Cauchy integral, namely,

lim_{t→0^{+}} C(f)(z − tν(z)) − lim_{t→0^{+}} C(f)(z + tν(z)) = f(z), z ∈ ∂Ω.

Since ψ defined by (2.9), (2.12) and (2.13) follow.

To prove the converse, one can simply reverse the argument above. This completes the proof. □

For a ∈ L^{p}(Ω), define T a(z) := \overline{∂}^{−1}ā + ∂^{−1}a as before. Since \overline{∂}^{−1} and ∂^{−1} are bounded from L^{p}(Ω) into C^{α}(\overline{Ω}) (α = \frac{p−2}{p}), there exists a constant C = C(Ω, p) such that

∥T a∥_{C^{α}(\overline{Ω})} ≤ C∥a∥_{p}.

Therefore, there are constants C and C_{1} depending only on Ω and p such that

(2.15)  ∥ exp(T a)∥_{∞} ≤ exp(C∥a∥_{p})
(2.16)  ∥ exp(T a)∥_{C^{α}(\overline{Ω})} ≤ C_{1}∥a∥_{p} exp(C∥a∥_{p}).

In fact, (2.16) follows from the following estimate:

| exp(T a)(z) − exp(T a)(ζ)| = | exp(T a)(z)||1 − exp[(T a)(ζ) − (T a)(z)]|
≤ C_{1} exp(C∥a∥_{p})∥a∥_{p}|(T a)(ζ) − (T a)(z)|
≤ C_{1} exp(C∥a∥_{p})∥a∥_{p}|z − ζ|^{α}.

Lemma 2.2. There exist constants C_{1} and C_{2} depending only on Ω and p such that if C_{1} exp(C_{2}∥a∥_{p})∥q∥_{p} < 1, then I − D_{k}^{−1}Q : L^{∞}(Ω) → L^{∞}(Ω) is invertible.

Proof. Fix α = \frac{p−2}{p}. Since Q : L^{∞}(Ω) → L^{p}(Ω) and D_{k}^{−1} : L^{p}(Ω) → C^{α}(\overline{Ω}) are bounded, D_{k}^{−1}Q is a compact operator on L^{∞}(Ω). Thus it suffices to establish the injectivity of I − D_{k}^{−1}Q. Suppose that u ∈ L^{∞}(Ω) satisfies

u − D_{k}^{−1}Qu = 0 in Ω.

Then,

E_{k}u − D^{−1}E_{k}Qu = 0

Considering the first column of the last equation, we have

(2.17)  u_{11} = \overline{∂}^{−1}(e^{T a}qu_{21})
(2.18)  e_{k}(z)u_{21} = ∂^{−1}(e_{k}e^{−T a}u_{11}).

It then follows from (2.15) and (2.17) that

∥u_{11}∥_{∞} ≤ C_{1}∥e^{T a}qu_{21}∥_{p}
≤ C_{1} exp(C_{2}∥a∥_{p})∥q∥_{p}∥u_{21}∥_{∞}.

We also have from (2.18) that

∥u_{21}∥_{∞} ≤ C_{3}∥e^{−T a}u_{11}∥_{∞}
≤ C_{3} exp(C_{2}∥a∥_{p})∥u_{11}∥_{∞}.

Note that all the constants in the estimates depend only on Ω and p . Thus if C_{1}C_{3} exp(2C_{2}∥a∥_{p})∥q∥_{p} < 1, then we have u_{21} = u_{11} = 0. In the same way, one can show that u_{12} = u_{22} = 0. This completes the proof. □

3. The Scattering Transform

We assume that I − D_{k}^{−1}Q : L^{∞}(Ω) → L^{∞}(Ω) is invertible and define

(3.1)  m(z, k) := (I − D_{k}^{−1}Q)^{−1}(I),

and the scattering transform by

(3.2)  S(k) = −\frac{1}{π} J ∫_{Ω} E_{k}(Q(z)m(z, k))dμ(z)

where the operator J is defined by

J ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) = ( \begin{matrix}0 & −ia_{12} \\ ia_{21} & 0 \end{matrix} ) .

Theorem 3.1. (i) For each fixed z, k → m(z, k) is differentiable as a function of k and satisfies

(3.3)  \frac{∂}{∂k̄}m(z, k) = m(z, k̄)Λ_{k}(z)S(k)

(ii) There are constants C_{3} and C_{4} such that

(3.4)  ∥m_{11}(·, k) − 1∥_{∞} + ∥m_{22}(·, k) − 1∥_{∞} ≤ \frac{C_{3}}{|k|}(∥a∥_{p} + 1) exp(C_{4}∥a∥_{p})∥q∥_{p},
(3.5)  ∥m_{12}(·, k)∥_{∞} + ∥m_{21}(·, k)∥_{∞} ≤ \frac{C_{3}}{|k|}(∥a∥_{p} + 1) exp(C_{4}∥a∥_{p}).

Proof. (i) is proved in [3].

Note that m satisfies

E_{k}m − D^{−1}E_{k}Qm = I.

Thus m_{11} and m_{21} satisfy

(3.6)  m_{11} = \overline{∂}^{−1}(e^{T a}qm_{21}) + 1
(3.7)  e_{k}(z)m_{21} = ∂^{−1}(e_{k}e^{−T a}m_{11}).

From (3.6), we have

∥m_{11} − 1∥_{C^{α}(\overline{Ω})} ≤ C_{1}∥e^{T a}qm_{21}∥_{p}

≤ C_{1} exp(C_{2}∥a∥_{p})∥q∥_{p}∥m_{21}∥_{∞}. (3.8)

Since ∂\overline{∂}^{−1} is a Calderón-Zygmund operator and bounded on L^{p} (1 < p < ∞), we also have from (3.6)

∥∂m_{11}∥_{p} ≤ C_{3}∥e^{T a}qm_{21}∥_{p}

≤ C_{3} exp(C_{2}∥a∥_{p})∥q∥_{p}∥m_{21}∥_{∞}. (3.9)

10 HYEONBAE KANG AND GUNTHER UHLMANN

Since e_{k}(z) = \frac{1}{ik}∂e_{k}(z), one can see from an integration by parts that

∂^{−1}(e_{k}e^{−T a}m_{11})(z)

= \frac{1}{ik} [\overline{C}(e_{k}e^{−T a}m_{11})|_{∂Ω})(z) + ∂^{−1}(e_{k}(∂T a)e^{−T a}m_{11})(z) − ∂^{−1}(e_{k}e^{−T a}∂m_{11})(z)]

:= \frac{1}{ik} [I_{1} + I_{2} + I_{3}].

A standard estimate for the Cauchy transform, (2.16), and (3.8) give

∥I_{1}∥_{∞} ≤ C_{4}∥e^{−T a}m_{11}∥_{C^{α}(∂Ω)}
≤ C_{4}∥e^{−T a}∥_{C^{α}(\overline{Ω})}∥m_{11}∥_{C^{α}(\overline{Ω})}

≤ C_{5} exp(C_{2}∥a∥_{p})(∥a∥_{p}∥q∥_{p}∥m_{21}∥_{∞} + 1). (3.10) Since ∂T is bounded on L^{p} (p < ∞), we have

∥I_{2}∥_{∞} ≤ C_{6}∥(∂T a)e^{−T a}m_{11}∥_{p}
≤ C_{6}∥∂T a∥_{p}∥e^{−T a}∥_{∞}∥m_{11}∥_{∞}

≤ C_{7}∥a∥_{p} exp(C_{2}∥a∥_{p})(∥q∥_{p}∥m_{21}∥_{∞} + 1). (3.11) From (3.9), we get

∥I_{3}∥_{∞} ≤ C∥e^{−T a}∥_{∞}∥∂m_{11}∥_{p}

≤ C_{1} exp(C_{2}∥a∥_{p})∥q∥_{p}∥m_{21}∥_{∞}. (3.12) It thus follows from (3.7), (3.10), (3.11), and (3.12) that

∥m_{21}∥_{∞} ≤ \frac{C_{8}}{|k|}[∥a∥_{p} exp(C_{2}∥a∥_{p})∥q∥_{p}∥m_{21}∥_{∞} + (∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})]

and hence

∥m_{21}∥_{∞} ≤ \frac{C_{9}}{|k|}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})

if |k| is so large that C_{8}∥a∥_{p} exp(C_{2}∥a∥_{p})∥q∥_{p} < \frac{1}{2}|k|. We can easily conclude from (3.6) that

∥m_{11} − 1∥_{∞} ≤ \frac{C_{10}}{|k|} (∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})∥q∥_{p}.

Estimates for m_{12} and m_{22} can be derived in the same way. This completes the proof. □ Lemma 3.2. Suppose that q ∈ W^{1,p}(Ω). Let the scattering matrix

S(k) = ( \begin{matrix}0 & s_{1}(k) \\ s_{2}(k) & 0 \end{matrix} ) .

Then, there are constants C_{1} and C_{2} such that

(3.13)  |s_{1}(k)| ≤ \frac{C_{1}}{|k| + 1}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})∥q∥_{W^{1,p}},
(3.14)  |s_{2}(k)| ≤ \frac{C_{1}}{|k| + 1}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})(∥q∥_{p} + 1).

Proof. Let

m = I + m̃.

By (3.2), we have

s_{1}(k) = −\frac{1}{π} ∫_{Ω} ẽ_{k}(z)e^{T a(z)}q(z)(1 + m̃_{22}(z))dμ(z).

If |k| is small, it is easy to see, using (3.4), that

|s_{1}(k)| ≤ C_{1}∥a∥_{p} exp(C_{2}∥a∥_{p})∥q∥_{p}.

If |k| is large, we get from (3.4)

|∫_{Ω} ẽ_{k}(z)e^{T a(z)}q(z)m̃_{22}(z)dμ(z)| ≤ \frac{C_{1}}{|k|}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})∥q∥_{p}^{2}.

By an integration by parts, we can show that

|∫_{Ω} ẽ_{k}(z)e^{T a(z)}q(z)dμ(z)| ≤ \frac{C_{1}}{|k|}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})∥q∥_{W^{1,p}}.

This gives (3.13).

Since

s_{2}(k) = −\frac{1}{π} ∫_{Ω} e_{−k}(z)e^{−T a(z)}(1 + m̃_{11}(z))dμ(z),

(3.14) follows from (3.4) and the same argument. This completes the proof. □

4. Proof of Theorem 1.6

In this section we prove Theorem 1.6. We need the following lemma due to Sun (equation (3.44) in [10]).

Lemma 4.1. If C_{a_{1},q_{1}} = C_{a_{2},q_{2}}, then ∂^{−1}a_{1} = ∂^{−1}a_{2} on ∂Ω.

Z. Sun prove this lemma under the assumption that the DN maps are the same. However, exactly the same argument works with the assumption C_{a_{1},q_{1}} = C_{a_{2},q_{2}}. Also Sun’s proof works under the weaker regularity assumptions assumed in this paper.

Lemma 4.2. Suppose that C_{a_{1},q_{1}} = C_{a_{2},q_{2}}. If I − D_{k}^{−1}Q^{(1)} is invertible on L^{∞}(Ω), so is I − D_{k}^{−1}Q^{(2)}.

Proof. Suppose that

(I − D_{k}^{−1}Q^{(2)})m^{(2)} = 0 in Ω.

We then define ψ^{(2)} by (2.9). Then, by Lemma 2.1, ψ_{21}^{(2)} and ψ_{22}^{(2)} are solutions of the equation (\overline{∂} + ā_{2})(∂ − a_{2})u − q_{2}u = 0 in Ω and there are f_{ij} ∈ H(ℂ \ \overline{Ω}) ∩ (ℂ \ Ω) with f_{ij}(z) = O(|z|^{−1}) as |z| → ∞, i, j = 1, 2, satisfying (2.12) and (2.13). Since C_{a_{1},q_{1}} = C_{a_{2},q_{2}}, there are solutions, say u_{1} and u_{2}, of (\overline{∂} + ā_{1})(∂ − a_{1})u − q_{1}u = 0 in Ω such that

\begin{cases}u_{1}|_{∂Ω} & = e^{∂^{−1}a_{2}}e^{−iz̄k̄}f̄_{21}, \\ (∂ − a_{1})u_{1}|_{∂Ω} & = e^{−\overline{∂}^{−1}ā_{2}}e^{izk}f_{11}, \end{cases}

and

\begin{cases}u_{2}|_{∂Ω} & = e^{∂^{−1}a_{2}}e^{−iz̄k}f̄_{22}, \\ (∂ − a_{1})u_{2}|_{∂Ω} & = e^{−\overline{∂}^{−1}ā_{2}}e^{izk̄}f_{12}. \end{cases}

Define m^{(1)} by

( \begin{matrix}(∂ − a_{1})u_{1} & (∂ − a_{1})u_{2} \\ u_{1} & u_{2} \end{matrix} ) = ( \begin{matrix}e^{−\overline{∂}^{−1}ā_{1}} & 0 \\ 0 & e^{∂^{−1}a_{1}} \end{matrix} ) m^{(1)}(z, k) ( \begin{matrix}e^{izk} & 0 \\ 0 & e^{−iz̄k} \end{matrix} ) .

Since ∂^{−1}a_{1} = ∂^{−1}a_{2} on ∂Ω by Lemma 4.1, we have m_{11}^{(1)} = f_{11}, e_{k}m_{21}^{(1)} = f̄_{21}, ẽ_{−k}m_{12}^{(1)} = f_{12}, m_{22}^{(1)} = f̄_{22} on ∂Ω and hence C(E_{k}m^{(1)}|_{∂Ω}) = 0 in Ω. Therefore, by Lemma 2.1, we have

(I − D_{k}^{−1}Q^{(1)})m^{(1)} = 0 in Ω.

Since I −D_{k}^{−1}Q^{(1)} is invertible, we have m^{(1)} = 0 in Ω. Thus we conclude that f_{ij} = 0. It then follows from (2.12) and (2.13) that ψ_{2j}^{(2)} = ∂ψ_{2j}^{(2)} = 0 on ∂Ω, j = 1, 2. We then have from the unique continuation property of the Schrödinger equation with magnetic potential ([1]) that ψ_{2j} = 0 in Ω, and hence m^{(2)} = 0. By the Fredholm alternative, we have the invertability of I − D_{k}^{−1}Q^{(2)} on L^{∞}(Ω). This completes the proof. □

Lemma 4.3. If C_{a_{1},q_{1}} = C_{a_{2},q_{2}} and I − D_{k}^{−1}Q^{(1)} is invertible on L^{∞}(Ω), then the corresponding scattering matrices coincide, i.e., S^{(1)}(k) = S^{(2)}(k) for all k ∈ ℂ.

Proof. Let m^{(j)} = (I − D_{k}^{−1}Q^{(j)})^{−1}(I), j = 1, 2. Then the scattering matrix S^{(j)}(k) is given by

S^{(j)}(k) = −\frac{1}{π} J ∫_{Ω} E_{k}Qm^{(j)}dμ(z)

= −\frac{1}{π} J ∫_{Ω} DE_{k}m^{(j)}dμ(z)

= ( \begin{matrix}0 & \frac{1}{2πi} ∫_{∂Ω} ẽ_{−k}m_{12}^{(j)}dz \\ −\frac{1}{2πi} ∫_{∂Ω} e_{k}m_{21}^{(j)}dz̄ & 0 \end{matrix} ) .

Thus it suffices to prove that m^{(1)} = m^{(2)} on ∂Ω. This can be proved using (2.11). In fact, define ψ^{(j)}, j = 1, 2, according to (2.9), namely,

ψ^{(j)}(z, k) := ( \begin{matrix}e^{−\overline{∂}^{−1}ā_{j}} & 0 \\ 0 & e^{∂^{−1}a_{j}} \end{matrix} ) m^{(j)}(z, k) ( \begin{matrix}e^{izk} & 0 \\ 0 & e^{−iz̄k} \end{matrix} ) .

Then, for k = 1, 2 and j = 1, 2, ψ_{1k}^{(j)} and ψ_{2k}^{(j)} satisfy

(\overline{∂} + ā_{j})(∂ − a_{j})ψ_{2k}^{(j)} − q_{j}ψ_{2k}^{(j)} = 0, in Ω
(∂ − a_{j})ψ_{2k}^{(j)} = ψ_{1k}^{(j)}.

Since C_{a_{1},q_{1}} = C_{a_{2},q_{2}}, there exist solutions, say u_{1} and u_{2}, of the equation (\overline{∂} + ā_{2})(∂ − a_{2})v − q_{2}v = 0 in Ω such that

u_{k}|_{∂Ω} = ψ_{2k}^{(1)}|_{∂Ω},
(∂ − a_{2})u_{k}|_{∂Ω} = ψ_{1k}^{(1)}|_{∂Ω}.

Define m by

( \begin{matrix}(∂ − a_{2})u_{1} & (∂ − a_{2})u_{2} \\ u_{1} & u_{2} \end{matrix} ) = ( \begin{matrix}e^{−\overline{∂}^{−1}ā_{2}} & 0 \\ 0 & e^{∂^{−1}a_{2}} \end{matrix} ) m(z, k) ( \begin{matrix}e^{izk} & 0 \\ 0 & e^{−iz̄k} \end{matrix} ) .

Since ∂^{−1}a_{1} = ∂^{−1}a_{2} on ∂Ω, we have m(z, k) = m^{(1)}(z, k) for all z ∈ ∂Ω and k ∈ ℂ and hence

E_{k}^{−1}C(E_{k}m|_{∂Ω}) = E_{k}^{−1}C(E_{k}m^{(1)}|_{∂Ω}) = I in Ω.

It then follows from (2.11) that

(I − D_{k}^{−1}Q^{(2)})m = I = (I − D_{k}^{−1}Q^{(2)})m^{(2)} in Ω.

Since (I − D_{k}^{−1}Q^{(2)}) is invertible, we have m = m^{(2)}. In particular, m^{(1)} = m^{(2)} on ∂Ω. This completes the proof. □

We are now ready to prove Theorem 1.6.

Proof of Theorem 1.6. If ϵ is so small that C_{1} exp(C_{2}M)ϵ < 1, then by Lemma 2.2, (I −D_{k}^{−1}Q^{(1)}) is invertible on L^{∞}(Ω). Then, (I −D_{k}^{−1}Q^{(2)}) is also invertible on L^{∞}(Ω) by Lemma 4.2. Define m^{(j)}, j = 1, 2, by

(I − D_{k}^{−1}Q^{(j)})m^{(j)} = I in Ω,

and let S^{(j)}(k) be the corresponding scattering matrix. Then by Lemma 4.3, we have S^{(1)}(k) = S^{(2)}(k) for all k ∈ ℂ. Put S(k) := S^{(1)}(k) = S^{(2)}(k) and m(z, k) := m^{(1)}(z, k) − m^{(2)}(z, k). Then, by (3.3), m satisfies

(4.1)  \frac{∂}{∂k̄}m(z, k) = m(z, k̄)Λ_{k}(z)S(k), k ∈ ℂ.

Moreover, by (3.4) and (3.5), we have

(4.2)  ∥m(·, k)∥_{∞} ≤ \frac{C}{1 + |k|}, k ∈ ℂ.

We now show that m ≡ 0. In view of (4.1), we obtain

\frac{∂}{∂k̄}(m_{11} ± m_{21}) = (m_{12}(z, k̄) ± m_{22}(z, k̄))e_{−k}s_{2},
\frac{∂}{∂k̄}(m_{12} ± m_{22}) = (m_{11}(z, k̄) ± m_{21}(z, k̄))ẽ_{k}s_{1}.

We then obtain using (4.2) that

m_{11} ± m_{21} = \overline{∂}_{k}^{−1}((m_{12}(z, k̄) ± m_{22}(z, k̄))e_{−k}s_{2}),
m_{12} ± m_{22} = \overline{∂}_{k}^{−1}((m_{11}(z, k̄) ± m_{21}(z, k̄))ẽ_{k}s_{1}).

Here \overline{∂}_{k}^{−1} is defined to be

\overline{∂}_{k}^{−1}f(w) := −\frac{1}{π} ∫_{ℂ} \frac{f(k)}{k − w}dμ(k), w ∈ ℂ,

and ∂_{k}^{−1} is defined likewise. It is well-known that \overline{∂}^{−1} : L_{−δ}^{2} → L_{−δ+1}^{2} is bounded where L_{δ}^{2} is the L^{2} space on ℂ weighted by (1 + |k|^{2})^{δ}. Thus it follows from (3.13) and (3.14) that

∥m_{11} ± m_{21}∥_{L_{δ}^{2}} ≤ C_{1}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})(∥q∥_{p} + 1)∥m_{12} ± m_{22}∥_{L_{δ}^{2}},
∥m_{12} ± m_{22}∥_{L_{δ}^{2}} ≤ C_{1}(∥a∥_{p} + 1) exp(C_{2}∥a∥_{p})∥q∥_{W^{1,p}}∥m_{11} ± m_{21}∥_{L_{δ}^{2}}.

Here the L_{δ}^{2}-norms are in the k-variable. Thus we have

∥m_{11} ± m_{21}∥_{L_{δ}^{2}} ≤ C(M + 1)^{2}e^{CM}ϵ∥m_{11} ± m_{21}∥_{L_{δ}^{2}}

for some constant C. If ϵ is so small that C(M + 1)^{2}e^{CM}ϵ < 1, then we have m_{11} ± m_{21} = 0 and m_{12} ± m_{22} = 0. Hence m = 0, or m^{(1)} = m^{(2)}. It then follows that Q^{(1)} = Q^{(2)} in Ω, and hence e^{T a_{1}} = e^{T a_{2}} and q_{1} = q_{2} in Ω. This completes the proof. □

Theorem 1.1 immediately follows from Theorem 1.6 once we observe that

∑_{j=1}^{2} (\frac{1}{i} \frac{∂}{∂x_{j}} − A_{j})^{2} = (\overline{∂} + ā)(∂ − a) + rota

where a := \frac{1}{2}(A_{2} + iA_{1}).

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Hyeonbae Kang, School of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

  E-mail address: hkang@math.snu.ac.kr

Gunther Uhlmann, Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA

  E-mail address: gunther@math.washington.edu