Increasing stability of the inverse boundary value problem for the Schr¨ odinger equation

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Increasing stability of the inverse boundary value problem for the Schr¨ odinger equation

V Isakov, S Nagayasu, G Uhlmann, and J-N Wang

Abstract. In this work we study the phenomenon of increasing stability in the inverse boundary value problem for the Schr¨odinger equation. This problem was previously considered by Isakov in which he discussed the phenomenon in different ranges of the wave number (or energy). The main contribution of this work is to provide a unified and easier approach to the same problem based on the complex geometrical optics solutions.

1. Introduction

Most of inverse problems are known to be severely ill-posed. This weakness makes it extremely difficult to design reliable reconstruction algorithms in prac- tice. However, in some cases, it has been observed numerically that the stability increases with respect to some parameter such as the wave number (or energy) (see, for example, [4] for the inverse obstacle scattering problem). Several rigorous justifications of the increasing stability phenomena in different settings were obtained by Isakov et al [7, 9, 10, 1, 2]. In particular, in [10], Isakov considered the Helmholtz equation with a potential

(1.1) ∆ + k²+ q(x)u(x) = 0 in Ω ⊂ Rⁿ

with n ≥ 3. He obtained stability estimates of determining q by the Dirichlet-to- Neumann map for different ranges of k, which demonstrate the increasing stability phenomena in k. The purpose of this work is to provide a more straightforward way to derive a similar estimate for the inverse boundary value for (1.1). In [10], Isakov used real geometrical optics solutions for the large wave number k. In this work,

2010 Mathematics Subject Classification. Primary 35R30; Secondary 65N21.

Key words and phrases. Stability estimate, Inverse boundary value problem, Schr¨odinger equation, Complex geometrical optics solutions.

Isakov was in part supported by the NSF grant DMS 10-08902 and by Emylou Keith and Betty Dutcher Distinguished Professorship at WSU.

Nagayasu was partly supported by Grant-in-Aid for Young Scientists (B).

Uhlmann was partly supported by NSF and The Fondation de Sciences Math´ematiques de Paris.

Wang was partly supported by the NSC grant 99-2115-M-002-006-MY3.

Isakov, Nagayasu, and Wang would like to dedicate this work to Gunther Uhlmann for his 60th birthday.

c

0000 (copyright holder) 1

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by more careful choice of an additional large parameter and a priori constraints we are able to use complex geometrical optics (CGO) solutions introduced by Calder´on [3] and Sylvester-Uhlmann [12] for all k ≥ 1. This will simplify the proof in [10].

Recently similar results were obtained by Isaev and Novikov [8] by using less explicit and more complicated methods of scattering theory.

In this work, instead of considering the Dirichlet-to-Neumann map, we define the boundary measurements to be the Cauchy data corresponding to (1.1)

Cq =

u|∂Ω,∂u

∂ν _∂Ω

, where u is a solution to (1.1)

.

Hereafter, ν is the unit outer normal vector of ∂Ω. Assume that Cq₁ and Cq₂ are two Cauchy data associated with refraction indices q1and q2, respectively. To measure the distance between two Cauchy data, we define

dist(Cq₁, Cq₂) = max (

max

(f,g)∈C_q1 min

( ef ,eg)∈C_q2

k(f, g) − ( ef ,eg)k_H1/2⊕H^−1/2

k(f, g)k_H1/2⊕H^−1/2

,

max

(f,g)∈C_q2

min

( ef ,eg)∈C_q1

k(f, g) − ( ef ,eg)k_H1/2⊕H^−1/2

k(f, g)k_H1/2⊕H^−1/2

) , where

k(f, g)k_H1/2⊕H^−1/2= kf k²_H1/2(∂Ω)+ kgk²_H−1/2(∂Ω)

^1/2 . Our main theorem is stated as follows.

Theorem 1.1. Let n ≥ 3. Assume C^q1 and Cq₂ are Cauchy data corresponding to q1(x) and q2(x), respectively. Let s > n/2 and M > 0. Assume kqlk_Hs(Ω)≤ M (l = 1, 2) and supp (q1− q2) ⊂ Ω. Denoteq the zero extension of qe 1− q2. Then for k ≥ 1 and dist(Cq₁, Cq₂) ≤ 1/e we have the following stability estimate:

(1.2) kqke_H−s(Rⁿ)≤ Ck⁴dist(Cq₁, Cq₂) + C

k + log 1 dist(C_q₁, C_q₂)

−(2s−n)

, where C > 0 depends only on n, s, Ω, M and supp (q1− q2).

From estimate (1.2), it is obvious that the stability behaves more like Lipschitz type when k is large. We would like to point out that unlike in the acoustic case where the constant associated with the Lipschitz estimate grows exponentially in k [11], the constant here grows only polynomially in k. Similarly, the corresponding constant obtained in [10] (see estimate (8) there) also grows polynomially in k.

The paper is organized as follows. In Section 2, we will collect some known results about the CGO solutions and an estimate for the difference of potentials, which are essential tools in the proof. In Section 3, we present a detailed proof of Theorem 1.1.

2. Preliminaries

To begin, we state the existence of CGO solutions for (1.1). These special solutions are first constructed by Sylvester and Uhlmann [12]. Another construction based on the Fourier series is given by H¨ahner [6].

Lemma 2.1. Let s > n/2. Assume that ζ = η + iξ (η, ξ ∈ Rⁿ) satisfies

|η|²= k²+ |ξ|² and η · ξ = 0,

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i.e., ζ · ζ = k². Then there exist constants C∗ and C > 0, which are independent of k, such that if |ξ| > C∗kqkH^s(Ω) then there exists a solution u to the equation (1.1) of the form

(2.1) u(x) = e^iζ·x 1 + ψ(x),

where ψ has the estimate

kψk_Hs(Ω)≤ C

|ξ|kqk_Hs(Ω).

Remark 2.2. Note that the correction term ψ decays in Im ζ. This property is crucial in obtaining that the constant associated with the Lipschiz estimate grows only polynomially in k.

Next inequality is an easy consequence of Alessandrini’s identity. We refer to [5] for the proof.

Proposition 2.3. Let ul and C_q_l be solution and Cauchy data to the equation (1.1) with q = ql, respectively (l = 1, 2). Then the following estimate holds:

Z

Ω

(q2− q1)u1u2dx

≤

u1,∂u₁

∂ν

_H_1/2_⊕H−1/2

u2,∂u₂

∂ν

_H_1/2_⊕H−1/2

dist(Cq1, Cq1).

3. Proof of main theorem To prove Theorem 1.1, we first derive two lemmas.

Lemma 3.1. Under the assumptions in Theorem 1.1, (3.1) |Fq(rω)| ≤ Cke ⁴e^Cadist(Cq₁, Cq₂) +C

akqke _H^−s_(Rn)

holds for k ≥ 1, r ≥ 0, ω ∈ Rⁿ with |ω| = 1 and a > C_∗M with k²+ a² > r²/4, where C > 0 depends only on n, s, M, Ω and supp (q1− q2) and C_∗ is the constant given in Lemma 2.1.

Proof. We will use CGO solutions (2.1) with appropriately chosen parameter ζ. Let us denote ζl= ηl+ iξl, l = 1, 2. We can choose ω^⊥,ωe^⊥ ∈ Rⁿ satisfying

ω · ω^⊥= ω ·eω^⊥= ω^⊥·ωe^⊥= 0 and |ω^⊥| = |ωe^⊥| = 1.

Now we set

ξ1= aω^⊥, η1= −r 2ω +

r

k²+ a²−r² 4 ωe^⊥, ξ₂= −ξ₁ and η₂= −rω − η₁,

and thus

ξ_l· η_l= 0, |η_l|²= k²+ |ξ_l|²

and |ξl| = a ≥ C∗M ≥ C∗kq`k_Hs(Ω). From Lemma 2.1, there exist CGO solutions ul(x) = e^iζ^l^x 1 + ψl(x)

to equation (1.1) with q = q_l, where ψ_lsatisfies kψlkH^s(Ω)≤ C

|ξl|kqlkH^s(Ω).

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Note that ψl also satisfies the estimate (3.2) kψlkH^s(Ω)≤ C

|ξl|kqlkH^s(Ω)≤ CM

a < CM C∗M = C

C∗

.

Now, by Proposition 2.3 and using the relation −rω = ζ₁+ ζ₂, we have that

Z

Ωeq(x)e^−irω·x(1 + ψ₁)(1 + ψ₂) dx

= Z

Ω

(q₂− q₁)u₁u₂dx

≤

u1,∂u1

∂ν

_H_1/2

⊕H^−1/2

u2,∂u2

∂ν

_H_1/2

⊕H^−1/2

dist(Cq₁, Cq₁).

Subsequently, we obtain

|Feq(rω)| = Z

Ωeq(x)e^−irω·xdx (3.3)

≤ Z

Ωq(x)ee ^−irω·x(1 + ψ1)(1 + ψ2) dx +

Z

Ωq(x)ee ^−irω·x(ψ1+ ψ2+ ψ1ψ2) dx

≤

u1,∂u1

∂ν

_H_1/2_⊕H−1/2

u2,∂u2

∂ν

_H_1/2_⊕H−1/2

dist(Cq₁, Cq₁)

+ Z

Ωq(x)ee ^−irω·x(ψ1+ ψ2+ ψ1ψ2) dx . In view of (3.3), we want to estimate

(ul, ∂ul/∂ν)

_H_1/2_⊕H_−1/2. Recall that ul solves (1.1) with q = ql. Using assumptions kqlk_Hs(Ω)≤ M , and s > n/2, and k ≥ 1, we have that

∂ul

∂ν

_H_−1/2_(∂Ω)

≤ Ck²kulk_L2(Ω)+ Ck∇ulk_L2(Ω)

and thus

u_l,∂u_l

∂ν

_H_1/2

⊕H^−1/2

≤ Ck²kulkL²(Ω)+ Ck∇u_lkL²(Ω). We now choose R₀> 0 large enough such that Ω ⊂ B_R₀(0). Then we have

|ul(x)| ≤ e^−ξ^l^·x 1 + |ψl(x)| ≤ Ce^|ξ^l^|R⁰ = Ce^aR⁰ since

|ψl(x)| ≤ kψlk_L∞(Ω)≤ Ckψlk_Hs(Ω)≤ C by s > n/2 and (3.2). It follows that

kulkL²(Ω)≤ Ce^aR⁰.

On the other hand, in view of k∇ψ_lkL²(Ω)≤ kψlkH^s(Ω) ≤ C (s > n/2 ≥ 3/2 > 1) and (3.2), we can estimate

k∇ulk_L2(Ω)=

iulζl+ e^iζ^l^·•∇ψl

_L₂_(Ω)≤ |ζl|kulk_L2(Ω)+ e^|ξ^l^|R⁰k∇ψlk_L2(Ω)

≤ C k + |ξ_l|eâR⁰+ Ce^|ξ^l^|R⁰ = C(k + a)eâR⁰+ CeâR⁰ ≤ Cke^Ca.

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Summing up, we obtain

ul,∂ul

∂ν

_H_1/2_⊕H−1/2

≤ Ck²kulkL²(Ω)+ Ck∇ulkL²(Ω)

(3.4)

≤ Ck²e^Ca+ Cke^Ca≤ Ck²e^Ca. Note that here C depends on n, s, M , and the diameter of Ω.

Let χ ∈ C₀^∞(Ω) be a cut-off function satisfying χ ≡ 1 near supp (q1− q2), then we have

Z

Ωq(x)ee ^−irω·x(ψ1+ ψ2+ ψ1ψ2) dx (3.5)

= Z

Ωeq(x)χ(x)e^−irω·x(ψ1+ ψ2+ ψ1ψ2) dx

≤ Z

Ω

|q(x)||χ(ψe ₁+ ψ₂+ ψ₁ψ₂)| dx

≤ kqke _H−s(Ω)kχ(ψ₁+ ψ₂+ ψ₁ψ₂)k_Hs(Ω). Since s > n/2 and (3.2), we can estimate

kχ(ψ1+ ψ2+ ψ1ψ2)k_Hs(Ω)

(3.6)

≤ kχkH^s(Ω) kψ1kH^s(Ω)+ kψ2kH^s(Ω)+ kψ1kH^s(Ω)kψ2kH^s(Ω)

≤ kχkH^s(Ω)

CM a +CM

a + C C∗

·CM a

≤ C a.

Finally, (3.1) follows from (3.3), (3.4), (3.5), and (3.6).

The following lemma is an easy corollary of Lemma 3.1.

Lemma 3.2. Suppose that the assumptions in Theorem 1.1 hold. Let R > C∗M with C_∗being the constant given in Lemma 2.1. Then for k ≥ 1, r ≥ 0 and ω ∈ Rⁿ with |ω| = 1, the following estimates hold true: if 0 ≤ r ≤ k + R then

(3.7) |Fq(rω)| ≤ Cke ⁴e^CRdist(Cq₁, Cq₂) +C

Rkqke _H^−s_(Rn); if r ≥ k + R then

(3.8) |Fq(rω)| ≤ Cke ⁴e^Crdist(C_q₁, C_q₂) +C

rkeqk_H−s(Rⁿ).

Proof. It is enough to take a = R when 0 ≤ r ≤ k + R, and take a = r when

r ≥ k + R in Lemma 3.1.

Now we prove our main theorem.

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Proof of Theorem 1.1. Written in polar coordinates, we have that kqke ²_H−s(Rⁿ)= C

Z ∞ 0

Z

|ω|=1

|Fq(rω)|e ²(1 + r²)^−srⁿ⁻¹dωdr (3.9)

= C

Z k+R 0

Z

|ω|=1

|Fq(rω)|e ²(1 + r²)^−srⁿ⁻¹dωdr

+ Z T

k+R

Z

|ω|=1

+ Z ∞

T

Z

|ω|=1

=: C(I₁+ I₂+ I₃),

where R > C_∗M and T ≥ k + R are parameters which will be chosen later.

Our task now is to estimate each integral separately. We begin with I3. Since

|Fq(rω)| ≤ Ckqe 1− q2k_L2(Ω), q1− q2∈ H₀^s(Ω) and s > n/2, we get

I₃≤ C Z ∞

T

kq1− q2k²_L2(Ω)(1 + r²)^−srⁿ⁻¹dr ≤ CT^−mkq1− q2k²_L2(Ω)

(3.10)

≤ CT^−m

εkq1− q2k²_H−s(Ω)+1

εkq1− q2k²_Hs(Ω)

≤ CT^−m

εkqke ²_H−s(Rⁿ)+1 ε

for ε > 0, where m := 2s − n.

On the other hand, by estimate (3.7), we can obtain

I1≤ Z k+R

0

Ck⁴e^CRdist(Cq₁, Cq₂) +C

Rkqke _H^−s_(Rn)

2

(1 + r²)^−srⁿ⁻¹dr (3.11)

≤ C

k⁸e^CRdist(Cq1, Cq2)²+ 1

R²kqke ²_H−s(Rⁿ)

Z ∞ 0

(1 + r²)^−srⁿ⁻¹dr

= C

k⁸e^CRdist(C_q₁, C_q₂)²+ 1

.

In the same way, using estimate (3.8), we have

I₂≤ C Z T

k+R

Ck⁴e^Crdist(C_q₁, C_q₂) +C

rkeqk_H−s(Rⁿ)

2

(1 + r²)^−srⁿ⁻¹dr (3.12)

≤ Ck⁸dist(Cq₁, Cq₂)² Z T

k+R

e^Cr(1 + r²)^−srⁿ⁻¹dr

+ Ckeqk²_H−s(Rⁿ)

Z T k+R

(1 + r²)^−srⁿ⁻¹dr

≤ C

k⁸e^CTdist(C_q₁, C_q₂)²+ 1

,

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where we have used Z T

k+R

e^Cr(1 + r²)^−srⁿ⁻¹dr ≤ e^CT Z T

k+R

(1 + r²)^−srⁿ⁻¹dr

≤ e^Ct Z ∞

0

(1 + r²)^−srⁿ⁻¹dr = Ce^CT,

Z T k+R

(1 + r²)^−srⁿ⁻¹dr ≤ Z T

k+R

r^−2s+n−1dr

≤ 1

2s − n + 2

1

(k + R)^2s−n+2 ≤ C

(k + R)² ≤ C R², and s > n/2, k ≥ 1. Combining (3.9)–(3.12) gives

kqke ²_H−s(Rⁿ)≤ C(I1+ I2+ I3) (3.13)

≤ C

k⁸e^CRdist(C_q₁, C_q₂)²+ 1

+ C

k⁸e^CT dist(Cq₁, Cq₂)²+ 1

+ CT^−m

εkqke ²_H−s(Rⁿ)+1 ε

≤ C

2

R² + εT^−m

kqke ²_H−s(Rⁿ)+ Ck⁸e^CRdist(Cq₁, Cq₂)²

+ Ck⁸e^CT dist(Cq₁, Cq₂)²+CT^−m ε . To continue, we consider the following two cases:

(i) k + R ≤ p log 1

A and (ii) k + R ≥ p log 1 A,

where R > C_∗M and p > 0 are constants which will be determined later. We begin with the first case (i). Taking

(3.14) R > 2

√ C and ε = cT^m (c 1), we deduce that

(3.15) kqke ²_H−s(Rⁿ)≤ Ck⁸A + Ck⁸e^CTA + CT^−2m for any T ≥ k + R by (3.13), where A = dist(C_q₁, C_q₂)².

Now we choose T = p log(1/A), which is greater than or equal to k + R by the condition (i). Our current aim is to show that there exists C₁> 0 such that

(3.16) k⁸e^CTA ≤ C1

k + log 1 A

−2m

and

(3.17) T^−2m≤ C₁

k + log 1 A

−2m

.

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Substituting (3.16) and (3.17) into (3.15) clearly implies (1.2). We remark that (3.17) is equivalent to

(3.18) C₁^−1/2m

k + log 1 A

≤ p log 1 A. Since we have

k + log 1

A ≤ (k + R) + log 1

A ≤ (p + 1) log 1 A by (i), condition (3.18) (i.e. (3.17)) holds whenever

(3.19) C₁^−1/2m≤ p

p + 1. On the other hand, condition (3.16) is equivalent to (3.20) 8 log k + (Cp − 1) log 1

A + 2m log

k + log 1 A

≤ log C1. Using (i), we can bound the left-hand side of (3.20) by

(LHS of (3.20)) ≤ 8 log p + 2m log(p + 1) + (Cp − 1) log 1

A+ 2(m + 4) log log 1 A. Choosing

(3.21) p ≤ 1

2C, we can see that

(LHS of (3.20))

≤ 8 log 1

2C + 2m log

1 2C + 1

−1 2log 1

A+ 2(m + 4) log log 1 A

≤ 8 log 1

2C + 2m log

1 2C + 1

+ max

z≥2

−1

2z + 2(m + 4) log z

= 8 log 1

2C + 2m log

1 2C + 1

+ 2(m + 4) log(4m + 16) − 1.

Therefore, condition (3.20) (i.e. (3.16)) is satisfied provided (3.22) 8 log 1

2C + 2m log

1 2C + 1

+ 2(m + 4) log(4m + 16) − 1 ≤ log C1. Next we consider case (ii). We choose T = k + R and observe that the term I2

in (3.9) does not appear in this case. Hence, instead of (3.13), we have kqke ²_H−s(Rⁿ)

≤ C

1

R² + εT^−m

kqke ²_H−s(Rⁿ)+ Ck⁸e^CRdist(C_q₁, C_q₂)²+CT^−m ε Setting ε = T^m/R² implies that

kqke ²_H−s(Rⁿ)≤ 2C

R²kqke ²_H−s(Rⁿ)+ Ck⁸e^CRA + CR²(k + R)^−2m. Now we choose

(3.23) R > 2√

C and obtain that

keqk²_H−s(Rⁿ)≤ Ck⁸A + C(k + R)^−2m,

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which implies the desired estimate (1.2) since from condition (ii) we have k + R ≥ k

2 +k + R 2 ≥ k

2 +p 2log 1

A ≥ min{p, 1}

2

k + log 1 A

.

As the last step, we choose appropriate R, p, and C₁to complete the proof. We first pick R > C_∗M sufficiently large satisfying (3.14) and (3.23) and then choose p small enough satisfying (3.21). Finally, we take C1large enough satisfying (3.19)

and (3.22).

4. Conclusion

We think that increasing stability is an important feature of the inverse boundary problem for the Schr¨odinger potential which should lead to higher resolution of numerical algorithms. It is important to collect numerical evidence of this phenomenon. Our method is based on the CGO solutions constructed in [6] where the constants in Lemma 2.1 are explicit. So most likely one can give explicit constants in Theorem 1.1 at least for particular domains Ω like balls. Contrary to the acoustic case [11], the constants in the estimate (1.2) depend only polynomially on k. It is an important and challenging question to determine whether the exponential de- pendence on k of the estimates in [11] is indeed generic if there are no assumptions on rays.

References

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[6] P. H¨ahner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308.

[7] T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.

[8] M. Isaev and R. Novikov, Energy and regularity dependent stability estimates for the Gelfand’s inverse problem in multi dimensions, J. Inverse Ill-Posed Problems, 20 (2012), 313-325.

[9] V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control methods in PDE-dynamical systems, 255V267, Contemp. Math., 426, AMS, Providence, RI, 2007.

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Department of Mathematics and Statistics, Wichita State University, KS 67260- 0033, USA.

E-mail address: victor.isakov@wichita.edu

Department of Mathematical Sciences, University of Hyogo, Himeji 671-2201, Japan.

E-mail address: sei@sci.u-hyogo.ac.jp

Department of Mathematics, University of Washington, Box 354305, Seattle, WA 98195-4350, USA & Fondation de Sciences Math´ematiques de Paris

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

Department of Mathematics, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan.

E-mail address: jnwang@math.ntu.edu.tw