Time-Fractional Schrödinger Equation

• TFSE is defined by substituting the standard first-order time derivative with a fractional (Caputo) derivative, introducing temporal nonlocality and inherent memory effects.
• The equation encompasses multiple formulations—such as Riemann–Liouville, Hadamard–Caputo, and conformable derivatives—that critically influence spectral behavior and unitarity.
• It finds applications in numerical discretization for weakly singular initial layers and in modeling non-Markovian open quantum systems with implications for quantum transport and information processing.

The Time-Fractional Schrödinger Equation (TFSE) is a family of Schrödinger-type evolution equations in which the first-order time derivative is replaced by a fractional derivative. In the literature, this replacement appears in several inequivalent forms, including $i\,{}^{C}D_t^\alpha u+\Delta u+f(|u|^2)u=0$, $i h\,\partial_t^\alpha\psi=\hat H\psi$, and $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$, with $0<\alpha<1$ in the standard time-fractional setting and $1<\alpha<2$ in diffusion-wave-type variants (Ma et al., 14 Apr 2025, Iomin, 2011, Bayin, 2011, Su et al., 2019). Across these formulations, the common feature is temporal nonlocality: the state at time $t$ depends on its prior history through a memory kernel, typically of Caputo type.

1. Formal definitions and competing formulations

TFSEs are often introduced by direct analogy with fractional diffusion: one replaces the first-order time derivative by a fractional derivative of order $0<\alpha\le 1$. In Iomin’s formulation, the equation is written

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$

with the time-fractional derivative understood as a Caputo derivative,

$$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$

and all variables dimensionless for dimensional consistency (Iomin, 2011). Bayın adopts a related but distinct convention,

$$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$

with $i h\,\partial_t^\alpha\psi=\hat H\psi$0 and $i h\,\partial_t^\alpha\psi=\hat H\psi$1 as $i h\,\partial_t^\alpha\psi=\hat H\psi$2 (Bayin, 2011).

A broader space-time fractional framework couples a Caputo time derivative to the quantum Riesz-Feller space-fractional derivative. For time-independent potentials,

$i h\,\partial_t^\alpha\psi=\hat H\psi$3

with $i h\,\partial_t^\alpha\psi=\hat H\psi$4, $i h\,\partial_t^\alpha\psi=\hat H\psi$5, and $i h\,\partial_t^\alpha\psi=\hat H\psi$6. Separation of variables gives the pure time-fractional equation

$i h\,\partial_t^\alpha\psi=\hat H\psi$7

together with a stationary space-fractional eigenproblem (Baqer et al., 2017).

Other derivative choices materially alter the theory. The Hadamard–Caputo derivative is used in the Cauchy problem

$i h\,\partial_t^\alpha\psi=\hat H\psi$8

where the logarithmic kernel is adapted to multiplicative time scaling (Alotaibi et al., 2022). A conformable variant replaces the nonlocal Caputo operator by

$i h\,\partial_t^\alpha\psi=\hat H\psi$9

leading to equations of the form

$$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$0

in recent open-system studies (Wei et al., 2023). By contrast, a Riemann–Liouville-type derivative with lower bound $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$1 has been used to define

$$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$2

which is shown to be equivalent to the first-order equation

$$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$3

for a positive self-adjoint operator $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$4 (Rougirel et al., 2018).

2. Spectral solutions, Mittag–Leffler dynamics, and Green functions

For time-independent generators, Caputo-based TFSEs replace exponential phases by Mittag–Leffler time factors. Bayın shows that the separated time equation

$$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$5

has solution

$$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$6

and derives this both by Laplace transform and by Bromwich inversion with Fox $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$7-functions (Bayin, 2011). In the Hilbert-space setting, if $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$8 is positive self-adjoint and $$i^\alpha \hbar^\alpha\,{}^{C}D_t^\alpha\psi=\hat H_\alpha\psi$$9, the strong solution of

$0<\alpha<1$0

is

$0<\alpha<1$1

constructed by the spectral theorem and functional calculus (Górka et al., 2016).

In Fourier variables, the free-particle solution on $0<\alpha<1$2 takes the form

$0<\alpha<1$3

so the linear TFSE is governed by a Mittag–Leffler Fourier multiplier rather than the standard $0<\alpha<1$4 phase (Górka et al., 2016). For $0<\alpha<1$5, the higher-order equation

$0<\alpha<1$6

is represented by oscillatory kernels with symbols

$0<\alpha<1$7

which are used to derive Hölder regularity and pointwise convergence to initial data (Su et al., 2019).

The same special-function structure persists in scattering and potential problems. Baqer and Boyadjiev obtain free-particle and linear-potential solutions for the space-time fractional Schrödinger equation in terms of Fox $0<\alpha<1$8-functions, while also recovering the Mittag–Leffler time factor for the separated temporal ODE (Baqer et al., 2017). Dong constructs a fractional Green’s function for the time-dependent scattering problem, expresses it in Fox $0<\alpha<1$9-function form and in a computable series form, and derives an asymptotic scattered wave with correction of every order (Jianping, 2013).

3. Unitarity, probability conservation, and physical interpretation

A central issue in TFSE theory is that Caputo-based time-fractional evolution is generally not unitary in the standard Hilbert-space sense. Iomin emphasizes that, for the equation

$1<\alpha<2$0

the Green function is a Mittag–Leffler function and does not satisfy Stone’s theorem on one-parameter unitary groups; consequently, the evolution is not generated by a self-adjoint Hamiltonian in the usual sense (Iomin, 2011). In the exact quantum comb model, the $1<\alpha<2$1 TFSE emerges only after projection onto the zero Fourier component in the auxiliary coordinate, while the remaining Fourier sectors are not described by the reduced equation. Iomin therefore concludes that the naive substitution $1<\alpha<2$2 can discard essential dynamical information (Iomin, 2011).

Bayın’s analysis makes the non-unitary content explicit. For a normalized separable solution,

$1<\alpha<2$3

and the Mittag–Leffler time factor yields

$1<\alpha<2$4

for short times and

$1<\alpha<2$5

for long times, so the total probability is less than one and decays with time when $1<\alpha<2$6 (Bayin, 2011). Laskin’s time-fractional quantum framework reaches a related conclusion: there are no stationary states, the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time-fractional quantum system, and the norm is time-dependent for $1<\alpha<2$7 (Laskin, 2017).

Two distinct responses to this difficulty appear in the literature. One is structural reformulation: the Riemann–Liouville-type equation

$1<\alpha<2$8

with lower terminal $1<\alpha<2$9 is shown to be equivalent to

$t$0

so the evolution operator

$t$1

forms a strongly continuous unitary group and conserves the $t$2 norm (Rougirel et al., 2018). The other is reinterpretation by a dynamical metric: for a traceless non-Hermitian two-level system, the non-unitary Mittag–Leffler evolution operator obtained from the TFSE can be mapped into a unitary one by a time-dependent Dyson map and a time-dependent metric operator, so that the system evolves unitarily in a dynamical Hilbert space (Cius et al., 2022).

4. Nonlinear TFSEs, regularity theory, and nonexistence results

Nonlinear TFSEs have been studied in several analytically distinct forms. For a Hartree perturbation on $t$3,

$t$4

with $t$5, $t$6, $t$7, $t$8, and $t$9, local well-posedness is established in $0<\alpha\le 1$0. The mild solution is

$0<\alpha\le 1$1

and the data-to-solution map is continuous (Prado et al., 2019).

The existence theory is complemented by sharp nonexistence results in other fractional-time settings. For the Hadamard–Caputo equation

$0<\alpha\le 1$2

Alotaibi, Jleli, Ragusa, and Samet prove that no global weak solution exists under explicit sign conditions on the initial data. One sufficient criterion is

$0<\alpha\le 1$3

which is obtained by a test-function method adapted to the logarithmic structure of the Hadamard kernel (Alotaibi et al., 2022).

A different regime arises for $0<\alpha\le 1$4, where the equation

$0<\alpha\le 1$5

interpolates between Schrödinger and wave equations. In this setting, Hölder regularity is derived from asymptotics of oscillatory kernels and singular Fourier multipliers. The mild solution becomes a classical solution if

$0<\alpha\le 1$6

and one also obtains pointwise convergence to both $0<\alpha\le 1$7 and $0<\alpha\le 1$8 as $0<\alpha\le 1$9 (Su et al., 2019).

5. Numerical discretization and weakly singular initial layers

The most detailed recent numerical analysis in the provided literature is due to Ma, Sun, and Chen, who study the two-dimensional nonlinear TFSE

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$0

with homogeneous Dirichlet boundary conditions, $$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$1, and $$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$2 (Ma et al., 14 Apr 2025). The analysis is tailored to weakly singular solutions satisfying

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$3

so that $$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$4 as $$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$5.

Their fully discrete method combines the L1 approximation of the Caputo derivative,

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$6

the five-point difference operator for $$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$7, and a one-step lag for the nonlinearity,

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$8

The resulting scheme is

$$i h\,\partial_t^\alpha \psi(x,t)=\hat H(x)\psi(x,t),$$9

and each time step solves a sparse linear system with matrix $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$0 (Ma et al., 14 Apr 2025).

The key analytical results are unconditional stability and pointwise-in-time convergence without any restriction on the grid ratio $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$1. In discrete $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$2, the difference of two numerical solutions satisfies

$$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$3

while the error $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$4 obeys

$$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$5

This yields first-order temporal accuracy at any fixed positive time and second-order spatial accuracy, but also quantifies the deterioration near $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$6 due to the initial layer (Ma et al., 14 Apr 2025).

The numerical experiments corroborate the analysis. For a manufactured weakly singular solution, the final-time error behaves like $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$7, while the global-in-time maximum error behaves like $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$8. Tests with $$D_t^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t \frac{f'(\tau)}{(t-\tau)^\alpha}\,d\tau,$$9 and $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$0 confirm the unconditional character of the method. Algorithmically, the history term costs $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$1 operations per step in a naive implementation, so the total cost is $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$2, and no fast convolution or sum-of-exponentials acceleration is used (Ma et al., 14 Apr 2025).

6. Open quantum systems, transport, and quantum-information applications

In open-system applications, TFSEs are used as effective models of non-Markovian dynamics. For the resonant dissipative Jaynes–Cummings model, the Caputo-based TFSE

$$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$3

admits an exact solution for a single-qubit open system, and the corresponding Quantum Speed Limit (QSL) analysis shows that non-Markovian memory effects can accelerate the evolution and reduce the QSL time. The explicit control parameters are the fractional order $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$4, coupling strength $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$5, and photon number $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$6; smaller $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$7, larger $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$8, and larger $$i^{\alpha}\hbar^{\alpha}\,{}^{C}D_t^\alpha \psi(x,t)=\hat H_\alpha \psi(x,t),\qquad \hat H_\alpha=-D_\alpha \frac{\partial^2}{\partial x^2}+V(x),$$9 reduce $i h\,\partial_t^\alpha\psi=\hat H\psi$00 in the reported regime (Wei et al., 2023).

Subsequent comparative studies sharpened the distinction between formulations. In the resonantly dissipative qubit model, three popular TFSEs—Naber’s TFSE I, Naber’s TFSE II, and XGF’s TFSE—do not preserve the total probability of the reduced system at fractional order. Moreover, the latter two do not describe non-Markovian dynamics at all fractional orders, only on restricted intervals of $i h\,\partial_t^\alpha\psi=\hat H\psi$01. A newly proposed conformable-derivative TFSE,

$i h\,\partial_t^\alpha\psi=\hat H\psi$02

with $i h\,\partial_t^\alpha\psi=\hat H\psi$03, is reported to preserve total probability for all fractional orders and to display non-Markovian features throughout the time evolution in both one- and two-qubit open systems (Wei et al., 2023). In the later RDJC comparison, Wei’s TFSE is further reported to capture non-Markovian accelerated dynamical features over the entire fractional-order range and to offer a significant simulation advantage in computational efficiency relative to Naber’s TFSE (Wei et al., 18 Jun 2026).

TFSEs have also been applied outside cavity-QED models. In a dimeric arrangement of perylene-bisimide organic molecules with effective Hamiltonian

$i h\,\partial_t^\alpha\psi=\hat H\psi$04

the Caputo TFSE

$i h\,\partial_t^\alpha\psi=\hat H\psi$05

is used to study coherence, entanglement, and CHSH nonlocality. Smaller $i h\,\partial_t^\alpha\psi=\hat H\psi$06 slows the decay of coherence and entanglement and can catalyze their dynamical generation from partially coherent initial states (Chhieb et al., 6 May 2026).

A different transport application concerns the half-plane Landau Hamiltonian in a constant magnetic field. For the TFSE

$i h\,\partial_t^\alpha\psi=\hat H\psi$07

with Caputo time derivative and $i h\,\partial_t^\alpha\psi=\hat H\psi$08, the large-time edge current undergoes a sharp transition: for fixed $i h\,\partial_t^\alpha\psi=\hat H\psi$09, it grows exponentially when $i h\,\partial_t^\alpha\psi=\hat H\psi$10, is asymptotically constant when $i h\,\partial_t^\alpha\psi=\hat H\psi$11, and decays when $i h\,\partial_t^\alpha\psi=\hat H\psi$12. The mean square displacement in the longitudinal direction exhibits the same transition, with ballistic scaling on the critical line and sub-ballistic scaling in the decaying regime (Hislop et al., 2024).

Taken together, these results show that “the TFSE” is not a single equation but a technically heterogeneous class of models. The derivative choice, phase convention, and operator realization determine whether the equation behaves as a reduced non-Markovian model, a non-unitary effective dynamics, or a reformulated unitary evolution, and they also determine which analytical tools—Mittag–Leffler calculus, Fox $i h\,\partial_t^\alpha\psi=\hat H\psi$13-functions, spectral functional calculus, test-function methods, or history-dependent discretizations—are appropriate for its study.