Affiliation:
1. School of Mathematics , Hunan University , Changsha 410082, Hunan Province , P. R. China
Abstract
Abstract
In this article, we study uniqueness and nonuniqueness for internal potential reconstruction from one boundary measurement in quantum fields that is related to the steady state Schrödinger equation.
It is an extension of our recent work [On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement, J. Comput. Appl. Math.
381 (2021), Paper No. 113029].
Based on the theory of the ND map and modified Bessel function, the uniqueness theorem of the inverse problem in two-dimensional and three-dimensional core-shell structure is established, respectively.
When different internal potential and shape are considered, the nonuniqueness results are also proved.
We then present some numerical examples to illustrate our theoretical results.
Reference35 articles.
1. M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
Nat. Bureau Standards Appl. Math. Ser. 55,
U. S. Government Printing Office, Washington 1964, 803–819.
2. G. S. Alberti and M. Santacesaria,
Calderón’s inverse problem with a finite number of measurements,
Forum Math. Sigma 7 (2019), Paper No. e35.
3. H. Ammari and H. Kang,
Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory,
Appl. Math. Sci. 162,
Springer, New York, 2007.
4. H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near-cloaking. Part II: The Helmholtz equation,
Comm. Math. Phys. 317 (2013), no. 2, 485–502.
5. H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem,
Comm. Math. Phys. 317 (2013), no. 1, 253–266.