We consider an inverse scattering problems for the biharmonic Schrödinger operator Δ2 + A · ∇ + V in three dimensions. By the Helmholtz decomposition, we take A = ∇p + ∇ ×ψ. The main contributions of this work are twofold. First, we derive a stability estimate of determining the divergence-free part ∇ ×ψ of A by far-field data at multiple wavenumbers. As a consequence, we further derive a quantitative stability estimate of determining . Both the stability estimates improve as the upper bound of the wavenumber increases, which exhibit the phenomenon of increased stability. Second, we obtain the uniqueness of recovering both A and V by partial far-field data. The analysis employs scattering theory to obtain an analytic domain and an upper bound for the resolvent of the fourth order elliptic operator. Notice that due to an obstruction to uniqueness, the corresponding results do not hold in general for the Laplacian, i.e., Δ + A · ∇ + V. This can be explained by the fact that the resolvent of the biharmonic operator enjoys a faster decay estimate with respect to the wavenumber compared with the Laplacian.
Skip Nav Destination
Article navigation
June 2024
Research Article|
June 06 2024
Inverse scattering problems of the biharmonic Schrödinger operator with a first order perturbation
Xiang Xu
;
Xiang Xu
a)
(Conceptualization, Formal analysis)
School of Mathematical Sciences, Zhejiang University
, Hangzhou 310027, China
Search for other works by this author on:
Yue Zhao
Yue Zhao
b)
(Conceptualization, Formal analysis)
School of Mathematical Sciences, Zhejiang University
, Hangzhou 310027, China
b)Author to whom correspondence should be addressed: [email protected]. Present address: School of Mathematics and Statistics, and Key Lab NAA–MOE, Central China Normal University, Wuhan 430079, China.
Search for other works by this author on:
b)Author to whom correspondence should be addressed: [email protected]. Present address: School of Mathematics and Statistics, and Key Lab NAA–MOE, Central China Normal University, Wuhan 430079, China.
a)
E-mail: [email protected]
J. Math. Phys. 65, 062105 (2024)
Article history
Received:
February 07 2024
Accepted:
May 20 2024
Citation
Xiang Xu, Yue Zhao; Inverse scattering problems of the biharmonic Schrödinger operator with a first order perturbation. J. Math. Phys. 1 June 2024; 65 (6): 062105. https://doi.org/10.1063/5.0202903
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
123
Views
Citing articles via
Cascades of scales: Applications and mathematical methodologies
Luigi Delle Site, Rupert Klein, et al.
Related Content
Inverse scattering problems for perturbed bi-harmonic operator
AIP Conference Proceedings (October 2016)
Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation
J. Acoust. Soc. Am. (January 2017)
On the estimation of the eigenfunctions of biharmonic operator in closed domain
AIP Conference Proceedings (June 2018)
Construction of cubic Ball surface based on biharmonic partial differentiation equation
AIP Conference Proceedings (July 2014)
Fourth-order convergence of a compact scheme for the one-dimensional biharmonic equation
AIP Conference Proceedings (September 2012)