On the time-fractional Schr ̈odinger equation: theoretical analysis and numerical solution by matrix Mittag–Leffler functions✩

This paper presents an original analysis of a time-dependent Schr ̈odinger equation with fractional time derivative. After the discretization of the spatial operator the equation is re- formulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schr ̈odinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solutionis split into two or three terms (depending on the order of thefractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memoryand a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented.