Reconstructing the potential for the 1D Schrödinger equation from boundary measurements

Sergei Avdonin (1) , Victor Mikhaylov (2) , Karim Ramdani (3)

1 Department of Mathematics and Statistics [Fairbanks]
2 RAS - Russian Academy of Sciences [Moscow]
3 CORIDA - Robust control of infinite dimensional systems and applications

We consider the inverse problem of determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the boundary. We use the Boundary Control method to recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We apply the algorithm to the situation when only the finite number of eigenvalues are known and prove the convergence of the method.

Domaines: Equations aux dérivées partielles [math.AP]

Sergei Avdonin, Victor Mikhaylov, Karim Ramdani. Reconstructing the potential for the 1D Schrödinger equation from boundary measurements. IMA Journal of Mathematical Control and Information, 2014, 31 (1), pp.137-150. ⟨10.1093/imamci/dnt009⟩. ⟨hal-00804268⟩