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

Abstract We investigate Caputo time–fractional quantum dynamics generated by a $$\mathcal{P}\mathcal{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> –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 $$\mathcal{P}\mathcal{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> –symmetric, and the time evolution is governed by a Caputo time–fractional Schrödinger equation of order $$0&lt;\alpha \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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 $$N=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 $$\mathcal{P}\mathcal{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> –symmetric complex periodic lattices.