An inverse problem for the fractional Schrödinger equation in a magnetic field

Abstract

This paper shows global uniqueness in an inverse problem for a fractional magnetic Schrödinger equation (FMSE): an unknown electromagnetic field in a bounded domain is uniquely determined up to a natural gauge by infinitely many measurements of solutions taken in arbitrary open subsets of the exterior.The proof is based on Alessandrini's identity and the Runge approximation property, thus generalizing some previous works on the fractional Laplacian.Moreover, we show with a simple model that the FMSE relates to a long jump random walk with weights.

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

“…Fractional magnetic Schrödinger equation. Section 6 of this paper extends the study of the fractional magnetic Schrödinger equation (FMSE) begun in [14], expanding the uniqueness result for the related inverse problem to the cases when s ∈ R + \Z. The direct problem for the classical magnetic Schrödinger equation (MSE) consists in finding a function u satisfying…”

“…There have been many studies in this direction (see for instance [54,53,55]). In our work, we will build upon the results from [14] and generalize them to higher order. Thus, for us the direct problem for FMSE asks to find a function u which satisfies…”

“…In section 6, we will construct the fractional magnetic Laplacian based on the fractional gradient operator ∇ s . The fractional gradient is based on the framework laid down in [19,20], and has been studied in the papers [15,14]. One should keep in mind that for s > 1 the fractional gradient is a tensor of order s rather than a vector.…”

“…In the corresponding inverse problem, we assume to know the DN map Λ s A,q : H s (Ω e ) → (H s (Ω e )) * , and we wish to recover A, q in Ω. In the cases when s ∈ (0, 1), it has been shown that the pair A, q can only be recovered up to a natural gauge [14]. We generalize this result to the case s ∈ R + \ Z.…”

“…Unique continuation of the fractional Laplacian has been extensively studied and used to show uniqueness results for fractional Schrödinger equations [14,28,29,75]. One version was already proved by Riesz [29,71] and similar methods were used in [42] to show a UCP of Riesz potentials I α which can be seen as fractional Laplacians with negative exponents.…”