Inverse problems for Schrödinger equations with Yang–Mills potentials in domains with obstacles and the Aharonov–Bohm effect
ABSTRACT We study the inverse boundary value problems for the Schrödinger equations with Yang-Mills potentials in a bounded domain Ω 0 ⊂ R n containing finite number of smooth obstacles Ω j , 1 ≤ j ≤ r. We prove that the Dirichlet-to-Neumann opeartor on ∂Ω 0 determines 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 ∂Ω 0 .
- Communications in Partial Differential Equations 01/1991; 16(6-7):1183-1195. · 1.03 Impact Factor
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ABSTRACT: In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.Physical Review - PHYS REV X. 01/1959; 115(3):485-491.
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: firstname.lastname@example.org
February 1, 2008
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
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
j=1Ωj) with Yang-Mills potentials
u + V (x)u − k2u = 0
with the boundary conditions
1 ≤ j ≤ r,
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
n),V′(x) are called
∂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
where g0,∂Ω0is the restriction of g0 to ∂Ω0. We shall prove the following
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.
∂x, V(1)= g−1
0V g0 then
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
· 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:
j(x))2u(p)+ V(p)(x)u(p)= 0, p = 1,2,
in Ω × (0,T0) with zero initial conditions
u(p)(x,0) = u(p)
t(x,0) = 0
and the Dirichlet boundary conditions
u(p)??∂Ωj×(0,T0)= 0, 1 ≤ j ≤ r, u(p)??∂Ω0×(0,T0)= f(x′,t), p = 1,2.
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 = (∂
· 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),
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.
k(x))u(p)+ V(p)(x)u(p)(x,t) = 0,
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
p(x)?−1are metric tensors in Ω
(p),gp(x) = det?gjk
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
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
= ˙ γ(τ) · A(p)(x(τ))c(p)(τ,γ),
c(p)(0,γ) = Im, p = 1,2, 0 ≤ τ ≤ τ1,