Inverse problems for time-fractional Schrödinger equations

Abstract We investigate some inverse problems for time-fractional Schrödinger equations involving the Caputo derivative of fractional order α ∈ ( 0 , 1 ) in a bounded domain. Such problems arise in fractional quantum systems and include determining initial data, source terms, and fractional orders from partial measurements taken on spatial sets of positive Lebesgue measure. We prove refined uniqueness results by relaxing the regularity assumptions on the initial data and source terms. Then, we give uniqueness results for measurements on discrete or pointwise data sets under additional assumptions. Our analysis is based on relevant properties of Mittag–Leffler functions with an imaginary argument, including the boundedness and asymptotic properties. Finally, we present numerical examples to illustrate some of the theoretical results.