We investigate Caputo time–fractional quantum dynamics generated by a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{P}\mathcal{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>–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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{P}\mathcal{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>–symmetric, and the time evolution is governed by a Caputo time–fractional Schrödinger equation of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0<\alpha \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{P}\mathcal{T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>–symmetric complex periodic lattices.

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

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2026年5月6日

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Volkan ALA

摘要

We investigate Caputo time–fractional quantum dynamics generated by a $\mathcal{P}\mathcal{T}$ $P T$ –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 $\mathcal{P}\mathcal{T}$ $P T$ –symmetric, and the time evolution is governed by a Caputo time–fractional Schrödinger equation of order $0<\alpha \le 1$ $0 < α \leq 1$ 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 $\mathcal{P}\mathcal{T}$ $P T$ –symmetric complex periodic lattices.