Exponential instability in inverse problems for the Schrödinger equation

We give an overview of known results on stability and instability in two problems: the Gel’fand inverse boundary value problem and the inverse scattering problem(3D). In particular, we present our new instability estimates, see [M. Isaev, Exponential instability in the Gel’fand inverse problem on the energy intervals, J. Inverse Ill-Posed Probl., Vol. 19(3), 2011, 453-473] and [M. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Func. Analiz i Ego Prilozheniya(to appear), e-print: arXiv:1012.5526]. In addition, some general scheme for obtaining instability estimates in inverse problems is also considered. 1. The Gel’fand inverse problem Consider the Schrödinger equation −∆ψ + v(x)ψ = Eψ for x ∈ D, where D is an open bounded domain in Rd, d ≥ 2, ∂D ∈ C2, v ∈ L∞(D). Define the map Φ = Φ(E) as follows: Φ(ψ|∂D) = ∂ψ ∂ν |∂D. Here we assume also that E is not a Dirichlet eigenvalue for operator −∆ + v in D. The map Φ(E) is called the Dirichlet-to-Neumann map. Problem 1. Given the map Φ(E), find v. The map Φ(E) can be given at fixed E or on energy intervals, for example, as the Schwartz kernel of the corresponding integral operator. Theorem 1 (variation of the result of G. Alessandrini [A1988]). Let aforementioned conditions for Problem 1 hold, d ≥ 3, m > d, M > 0 and supp vi ⊂ D, ||vi||Cm(D) ≤ M , i = 1, 2. Then there is a constant C1 = C1(M,D,m) > 0 such that ||v1 − v2||L∞(D) ≤ C1 ( log(3 + ||Φ1 − Φ2|| )−α , (1) where α = (m− d)/m and Φ1, Φ2 denote DtN maps for v1, v2 respectively, ||Φ1 − Φ2|| = ||Φ1 − Φ2||L∞(∂D)→L∞(∂D). The analogous estimate for d = 2 was proved by R. Novikov, M. Santacesaria in [NS2010]. A principal disadvantage of Alessandrini-type estimates: α < 1 for any m > d, even if m is very great. • R. Novikov [N2011] showed that estimate (1) holds with α = m − d in the case of the Gel’fand-Calderon inverse problem (E = 0) and d ≥ 3. • M. Santacesaria [S2011] showed that estimate (1) holds with α = m−2 in the case of the Calderon inverse problem (E = 0 and v is a potentaial of conductivity type) and d = 2. • N. Mandache [M2001] showed that estimate (1) can not hold in the case of the Gel’fandCalderon inverse problem (E = 0) and d ≥ 2 – with α > 2m− d for real-valued potentials, – with α > m for complex-valued potentials. • M. Isaev [I2011] extended results of [M2001] to the case of general E as well as to the case of the energy intervals: 2. The inverse scattering problem(3D) Consider the Schrödinger equation −∆ψ + v(x)ψ = Eψ, x ∈ R3, where the potential v is real-valued, v ∈ L∞(R3), v(x) = O(|x|−3−ε), |x| → ∞, for some ε > 0. Then for any k ∈ R3 \ {0} the equation with E = k2 has a unique continuous solution ψ+(x, k), with asymptotics of the form ψ+(x, k) = eikx − 2π2 ei|k||x| |x| f ( k |k| , x |x| , |k| ) + o ( 1 |x| ) as |x| → ∞ ( uniformly in x |x| ) , where f( k |k|, ω, |k|) with fixed k is a continuous function of ω ∈ S 2. The function f(θ, ω, s) is refered to as the scattering amplitude. Problem 2. Given the scattering amplitude f(θ, ω, s), find v. The scattering amplitude f(θ, ω, s) can be given at fixed E = s2 or on the energy interval. Theorem 2 (P. Stefanov [S1990]). Letm > 3/2, s,M > 0, ρ ∈ (0, 1) , supp vi ⊂ B(0, ρ), vi ∈ L∞(D)∩Hm(R3), ||vi||L∞(D) ≤M , i = 1, 2. Then for some 0 < α < 1 there is the constant C2 = C2(M,ρ) such that ||v1 − v2||L∞(D) ≤ C2 ( log(1 + ||f1(·, ·, s)− f2(·, ·, s)||−1 σ1,σ2 )−α , (2) where σ1 = 3/2, σ2 = −1/2, f1, f2 are the scattering amplitudes for v1, v2 respectively. P. Stefanov used the following (sufficiently strong) norm for the scattering amplitude: ||f(·, ·, s)||σ1,σ2 =  ∑ j1,p1,j2,p2 ( 2j1 + 1 es )2(j1+σ1)(2j2 + 1 es )2(j2+σ2) |aj1p1j2p2(s)| 2 1/2 , where aj1p1j2p2(s), j1, j2 ≥ 0, |p1| ≤ j1, |p2| ≤ j2, denotes the expansion coefficients of f(θ, ω, s) in the basis of spherical harmonics {Y p1 j1 × Y p2 j2 }: