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arXiv:math/0505554v1 [math.AP] 26 May 2005

Inverse problems for Schr¨ odinger equations

with Yang-Mills potentials in domains with

obstacles and the Aharonov-Bohm effect.

G.Eskin,Department of Mathematics, UCLA,

Los Angeles, CA 90095-1555, USA. E-mail: eskin@math.ucla.edu

February 1, 2008

Abstract

We study the inverse boundary value problems for the Schr¨ odinger

equations with Yang-Mills potentials in a bounded domain Ω0⊂ Rn

containing finite number of smooth obstacles Ωj,1 ≤ j ≤ r. We prove

that the Dirichlet-to-Neumann opeartor on ∂Ω0determines the gauge

equivalence class of the Yang-Mills potentials. We also prove that the

metric tensor can be recovered up to a diffeomorphism that is identity

on ∂Ω0.

1 Introduction.

Let Ω0be a smooth bounded domain in Rn, diffeomorphic to a ball, n ≥ 2,

containing r smooth nonintersecting obstacles Ωj, 1 ≤ j ≤ r. Consider the

Schr¨ odinger equation in Ω = Ω0\ (∪r

?2

j=1Ωj) with Yang-Mills potentials

(1.1)

n

?

j=1

?

−i∂

∂xjIm+ Aj(x)

u + V (x)u − k2u = 0

with the boundary conditions

(1.2)

u??∂Ωj= 0,

1 ≤ j ≤ r,

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(1.3)

u|∂Ω0= f(x′),

where Aj(x),V (x),u(x) are m × m matrices, Im is the identity matrix in

Cm. Let G(Ω) be the gauge group of all smooth nonsingular matrices in Ω.

Potentials A(x) = (A1,...,An),V and A′(x) = (A′

gauge equivalent if there exists g(x) ∈ G(Ω) such that

A′(x) = g−1Ag − ig−1(x)∂g

1,...,A′

n),V′(x) are called

(1.4)

∂x, V′= g−1V g.

Let Λ be the Dirichlet-to-Neumann (D-to-N) operator on ∂Ω0, i.e.

Λf = (∂u

∂ν+ i(A · ν)u)??∂Ω0,

where ν = (ν1,...,νn) is the unit outward normal to ∂Ω0 and u(x) is the

solution of (1.1), (1.2), (1.3)). We assume that the Dirichlet problem (1.1),

(1.2), (1.3)) has a unique solution. We shall say that the D-to-N operators

Λ and Λ′are gauge equivalent if there exists g0∈ G(Ω) such that

Λ′= g0,∂Ω0Λg−1

0,∂Ω0,

where g0,∂Ω0is the restriction of g0 to ∂Ω0. We shall prove the following

theorem:

Theorem 1.1. Suppose that D-to-N operators Λ′and Λ corresponding to

potentials (A′,V′) and (A,V ) respectively are gauge equivalent for all k ∈

(k0− δ0,k0+ δ0), where k0> 0, δ0> 0. Then potentials (A′,V′) and (A,V )

are gauge equivalent too.

If we replace A′,V′by A(1)= g−1

Λ = Λ1where Λ1is the D-to-N operator corresponding to (A(1),V(1)). The

proof of Theorem 1.1 gives that if Λ = Λ1then (A,V ) and (A(1),V(1)) are

gauge equivalent with a gauge g ∈ G(Ω) such that g|∂Ω0= Im. We shall

denote the subgroup of G(Ω) consisting of g such that g(x)|∂Ω0= Im by

G0(Ω). In the case when Ω0contains no obstacles Theorem 1.1 was proven

in [E] for n ≥ 3 and in [E3] for n = 2. Note that the result of [E] is stronger

since it requires that Λ = Λ(1)for one value of k only. In the case n = 2 the

proof of Theorem 1.1 is simpler than that in [E3] since it does not rely on

the uniqueness of the inversion of the non-abelian Radon transform.

0A′g0− ig−1

0

∂g0

∂x, V(1)= g−1

0V g0 then

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We shall prove Theorem 1.1 in two steps. In §2 we shall prove that (A,V )

and (A(1),V(2)) are locally gauge equivalent using the reduction to the inverse

problem for the hyperbolic equations as in [B], [B1], [KKL], [KL], [E1], and

in §3 we shall prove the global gauge equivalence using the results of §2 and

of [E2]. Following Yang and Wu (see [WY]) one can describe the gauge

equivalence class of A = (A1,...,An). Fix a point x(0)∈ ∂Ω0and consider all

closed paths γ in Ω starting and ending at x(0). Let x = γ(τ), 0 ≤ τ ≤ τ0,

be a parametric equation of γ, γ(0) = γ(τ0) = x(0). Consider the Cauchy

problem for the system

∂

∂τc(τ,γ) =dγ(τ)

dτ

· A(γ(τ))c(τ,γ), c(0,γ) = Im.

By the definition the gauge phase factor c(γ,A) is c(τ0,γ). Therefore A

defines a map of the group of paths to GL(m,C). The image of this map is

a subgroup of GL(m,C) which is called the holonomy group of A (see [Va]).

It is easy to show (c.f. §3) that c(γ,A(1)) = c(γ,A(2)) for all closed paths γ iff

A(1)and A(2)are gauge equivalent in Ω. As it was shown by Aharonov and

Bohm [AB] the presence of distinct gauge equivalent classes of potentials can

be detected in an experiment and this phenomenon is called the Aharonov-

Bohm effect. In $ 4 we consider the recovery of the Riemannian metrics from

the D-to-N operator in domains with obstacles.

2Inverse problem for the hyperbolic system.

Consider two hyperbolic system:

(2.1)

L(p)u =∂2

∂t2u(p)+

n

?

j=1

(−i∂

∂xjIm+ A(p)

j(x))2u(p)+ V(p)(x)u(p)= 0, p = 1,2,

in Ω × (0,T0) with zero initial conditions

(2.2)

u(p)(x,0) = u(p)

t(x,0) = 0

and the Dirichlet boundary conditions

(2.3)

u(p)??∂Ωj×(0,T0)= 0, 1 ≤ j ≤ r, u(p)??∂Ω0×(0,T0)= f(x′,t), p = 1,2.

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Here Ω = Ω0\ (∪r

u(p)(x,t), p = 1,2, are smooth m × m matrices. As in §1 introduce D-to-N

operators Λ(p)f = (∂

j

· νj)u??∂Ω0×(0,T0), p = 1,2.

for (2.1) when T0 = ∞ determines the D-to-N operator for (1.1) for all k

except a discrete set, and vice versa.

We shall prove the following theorem:

j=1Ωj) is the same as in §1, A(p)

j(x), 1 ≤ j ≤ n, V(p)(x),

∂ν+ i?n

j=1A(p)

Making the Fourier transform in t one can show that the D-to-N operator

Theorem 2.1. Suppose Λ(1)= Λ(2)and T0> maxx∈Ωd(x,∂Ω0) where d(x,∂Ω0)

is the distance in Ω from x ∈ Ω to ∂Ω0. Then potentials A(1)

n, V1(x) and A(2)

(1.4) holds with g ∈ G0(Ω).

Note that Theorem 2.1 implies Theorem 1.1. We can consider a more

general than (2.1) equation when the Eucleadian metric is replaced by an

arbitrary Riemannian metric:

j(x),1 ≤ j ≤

j(x),1 ≤ j ≤ n, V(2)(x) are gauge equivalent in Ω, i.e.

∂2u(p)

∂t2

+

n

?

j,k=1

1

?gp(x)(−i∂

∂xjIm+ A(p)

j(x))

?

gp(x)gjk

p(x)(−i∂

∂xjIm

+A(p)

k(x))u(p)+ V(p)(x)u(p)(x,t) = 0,

(2.4)

where ?gjk

are the same as in (2.1), Ω(p)= Ω0\ Ω

subset of ∂Ω0and let 0 < T < T0be small. Denote by ∆(0,T) the intersec-

tion of the domain of influence of Γ with ∂Ω0× [0,T]. We assume that the

domain of influence of Γ does not intersect Ω′p× [0,T].

Lemma 2.1. Suppose Λ(1)= Λ(2)on ∆(0,T). There exist neighborhoods

U(p)⊂ Ω(p),p = 1,2, U(p)∩∂Ω0= Γ and the diffeomorphism ϕ : U(1)→ U(2)

such that ϕ|Γ= I and ?gjk

and ϕ ◦ A(2)

exists g(x) ∈ G0(U(1)), g(x) = I on Γ such that (1.4) holds in U(1).

The proof of Lemma 2.2 is the same as the proof of Lemma 2.1 in [E1].

One should replace only the inner products of the form?u(x,t)v(x,t)dxdt

by?Tr(uv∗)dxdt where v∗is the adjoint matrix to v(x,t). We do not assume

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p(x)?−1are metric tensors in Ω

(p),gp(x) = det?gjk

′

p, Ω′

p?−1, A(p)

j(x),V(p)(x)

p= ∪r

j=1Ωjp. Let Γ be an open

2? = ϕ ◦ ?gjk

1?. Moreover A(1)

j, 1 ≤ j ≤ n, V(1)

j,1 ≤ j ≤ n, ϕ ◦ V(2)are gauge equivalent in U(1), i.e. there

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that matrices A(p)

obtained by the BC-method (see[B], [KKL]). Extend ϕ−1from U2to Ω(2)in

such a way that ϕ = I on ∂Ω0and ϕ is a diffeomorphism of Ω(2)and˜Ω(2)=

ϕ−1(Ω(2)). Also extend g(x) from U1to˜Ω(2)so that g(x) ∈ G0(˜Ω2), g = I

on ∂Ω0. Then we get that˜L(2)= g ◦ ϕ ◦ L(2)= L(1)in U(1).

Lemma 2.2. Let L(1)and L(2)be the operators of the form (2.4) in Ω(p)=

Ω0\ Ω′p, p = 1,2. Let B ⊂ Ω(1)∩ Ω(2)be simply-connected, ∂B ∩ ∂Ω0= Γ

be open and connected, and Ω(p)\ B be smooth. Suppose L(2)= L(1)in

B and Λ(1)= Λ(2)on ∂Ω0× (0,T0) where Λ(p)are the D-to-N operators

corresponding to L(p), p = 1,2. Then˜Λ(1)=˜Λ(2)where˜Λ(p)are the D-to-N

operators corresponding to L(p)in the domains (Ω(p)\ B) × (δ,T0− δ), δ =

maxx∈Bd(x,∂Ω0)), d(x,∂Ω0) is the distance in B between x ∈ B and ∂Ω0,

˜Λ(p)are given on ∂(Ω0\ B) × (δ,T0− δ).

Therefore Lemma 2.2 reduces the inverse problem in Ω(p)× (0,T0) to

the inverse problem in a smaller domain (Ω(p)\ B) × (δ,T0− δ). Combining

Lemmas 2.1 and 2.2 we can prove that for any x(0)∈ Ω(1)there exist a simply-

connected domain B1⊂ Ω(1), x(0)∈ B1, a diffeomorphism ϕ of˜Ω(2)onto Ω(2),

ϕ = I on ∂Ω0, such that g ∈ G0(˜Ω(2)) such that˜L(2)def

B1. To prove the global gauge equivalence and global diffeomorphism in the

case when Ω(1)is not simply-connected we shall use some additional global

quantities determined by the D-to-N operator (c.f. [E2]).

j,V(p)are self-adjoint In the latter case Lemma 2.1 can be

= g ◦ ϕ ◦ L(2)= L(1)in

3Global gauge equivalence.

In this section we shall prove Theorem 2.1. Fix arbitrary point x(0)∈ ∂Ω0.

Let γ be a path in Ω starting at x(0)and ending at x(1)∈ Ω, γ(τ) = x(τ) is

the parametric equation of γ, 0 ≤ τ ≤ τ1, x(0)= x(0), x(1)= x(τ1). Denote

by c(p)(τ,γ), p = 1,2, the solution of the system of differential equations

(3.1)

i∂c(p)(τ,γ)

∂τ

= ˙ γ(τ) · A(p)(x(τ))c(p)(τ,γ),

where

(3.2)

c(p)(0,γ) = Im, p = 1,2, 0 ≤ τ ≤ τ1,

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