Caputo Time–Fractional Dynamics of a PT–Symmetric Complex Periodic Schrödinger Operator

We investigate Caputo time–fractional quantum dynamics generated by a PT P T –symmetric complex periodic Schrödinger operator on the real line. The spatial part is a one-dimensional Schrödinger operator with an extended complex trigonometric potential that is L –periodic and PT P T –symmetric, and the time evolution is governed by a Caputo time–fractional Schrödinger equation of order 0<α ≤ 1 0 < α ≤ 1 with time-independent Hamiltonian H . Within a Bloch–Floquet framework we express the solution in terms of Mittag–Leffler functions of the Bloch band energies and obtain a band-resolved spectral expansion. For the finite-band Heun polynomial states of degree $$N=2$$ N = 2 we derive explicit evolution formulas and identify a finite-dimensional quasi-exactly solvable subspace. The analysis provides an analytic framework for Caputo time–fractional quantum dynamics in PT P T –symmetric complex periodic lattices.