Fractional Schrödinger Equation

• Fractional Schrödinger equation is a generalization of the classical equation using non-integer order derivatives to capture anomalous transport and nonlocal quantum effects.
• It replaces standard operators with fractional Laplacian and Caputo derivatives, leading to modified tunneling rates and non-Markovian, dissipative dynamics.
• Robust numerical methods and experimental setups in optics and quantum systems validate its applications in simulating complex media and anomalous quantum behavior.

The fractional Schrödinger equation (FSE) is a generalization of the classical Schrödinger equation in which differential operators of non-integer order—typically the fractional Laplacian (spatial) and/or Caputo-type (temporal) derivatives—replace the standard second-order Laplacian and first-order time derivative. This framework rigorously describes quantum evolution on fractal or multiscale media, supports the modeling of anomalous transport (Lévy flights), and captures memory effects and nonlocality, with applications ranging from quantum physics to optics, condensed matter, and signal processing.

1. Mathematical Formulations and Fractional Derivatives

The canonical time-dependent fractional Schrödinger equation, in one spatial dimension and units $\hbar=1$, takes the form

$$i\frac{\partial \psi(x,t)}{\partial t} = D_\alpha\,(-\Delta)^{\alpha/2} \psi(x,t) + V(x)\psi(x,t),$$

where $0<\alpha\le2$ is the Lévy index, $D_\alpha$ a kinetic coefficient, $(-\Delta)^{\alpha/2}$ the Riesz fractional Laplacian, and $V(x)$ the potential energy (Longhi, 2015, Liu et al., 2022). The spatial operator acts via its Fourier representation: $$\mathcal{F}\left[(-\Delta)^{\alpha/2}\psi\right](k) = |k|^\alpha\,\widehat{\psi}(k),$$ or equivalently (in real space) through a singular integral as

$$(-\Delta)^{\alpha/2} \psi(x) = \frac{1}{2\pi} \iint_{-\infty}^{\infty} |p|^\alpha\,\psi(\xi)\,e^{ip(x-\xi)}\,dp\,d\xi.$$

For time-fractional dynamics, the standard derivative is replaced by the Caputo derivative of order $0<\beta\le1$: $$D_t^\beta f(t) = \frac{1}{\Gamma(1-\beta)} \int_0^t (t-\tau)^{-\beta} f'(\tau)\,d\tau,$$ yielding time–space fractional models (Górka et al., 2016, Baqer et al., 2017, Baqer et al., 2017, Gabrick et al., 2023, Iomin, 2021). Variants involve the quantum Riesz–Feller derivative, which introduces asymmetry via a skewness parameter $\theta$: $$\mathcal{F}\{D^\alpha_\theta\psi(x)\}(p) = -|p|^\alpha e^{i(\mathrm{sgn}(p) \theta\pi/2)} \Psi(p),$$ with $$\theta\in[-\min\{\alpha,2\!-\!\alpha\},+\min\{\alpha,2\!-\!\alpha\}]$$. The Jumarie fractional derivative and other nonlocal definitions have also been discussed for coarse-grained quantum systems (Banerjee et al., 2016).

2. Solution Frameworks and Special Cases

2.1. Spectral Formulation and Well-posedness

For abstract self-adjoint generators $A$ on Hilbert space, strong solutions of the time-fractional Cauchy problem

$$D_t^\alpha u(t) = (-i)^\alpha A u(t), \qquad u(0)=u_0, \qquad 0<\alpha<1$$

exist uniquely and are represented via the operator family

$$U_\alpha(t) = \int_{\sigma(A)} E_\alpha\bigl( (-i t)^\alpha \lambda \bigr)\,dE_\lambda,$$

where $E_\alpha(z)$ is the one-parameter Mittag–Leffler function (Górka et al., 2016). As $\alpha \to 1^-$, $U_\alpha(t)$ recovers the unitary group $e^{-itA}$, showing the FSE continuously interpolates between memoryless (Markovian, unitary) and non-Markovian (nonunitary, dissipative) quantum evolutions.

2.2. Exact and Special Function Solutions

For free particles, or specific potentials $V(x)$, the FSE admits solutions in terms of special functions. With zero potential and Caputo time-fractional derivative, the fundamental solution is expressed via the Fox $H$-function (Baqer et al., 2017, Baqer et al., 2017): $$G(x,t) = \frac{1}{\alpha |x|} H^{1,1}_{2,2} \left[\frac{|x|}{(C_\alpha t^\beta)^{1/\alpha}} \,\Bigg|\, \begin{matrix} (1,1), (1,\beta/\alpha) \ (1,1), (1,\beta/\alpha) \end{matrix}\right].$$ For linear or delta potentials, spatial eigenfunctions typically involve Fox $H$ or Fox $H^{m,n}_{p,q}$ functions, generalizing Airy functions and exponentials (Baqer et al., 2017, Baqer et al., 2017).

In fractional infinite square wells, the eigenfunctions and spectra deviate significantly from their integer-order analogs due to nonlocal boundary effects; Riesz fractional operators yield eigenvalues that interpolate between linear and quadratic dependence on quantum number, with eigenfunctions increasingly concentrated near the walls as $\alpha$ decreases (Herrmann, 2012, Banerjee et al., 2016).

3. Physical Interpretations and Properties

Fractional spatial derivatives encode Lévy flight path integrals, leading to algebraic (power-law) tails in eigenfunctions: $$\psi(x) \sim |x|^{-(\alpha+1)} \qquad (\text{outside potential wells}),$$ in contrast to the exponential decay of integer-order quantum mechanics (Lewis et al., 2024). This fundamentally modifies tunneling rates—yielding fractionally enhanced quantum tunneling—and affects the count and structure of bound states. For example, the tunneling transmission coefficient through a barrier of width $2a$ becomes

$$T_\alpha \propto \left(a\,\kappa^{1/\alpha}\right)^{-2(\alpha+1)/\alpha},$$

which grows substantially compared to the exponential suppression for $\alpha=2$ (Lewis et al., 2024).

Time-fractional derivatives impart memory to quantum dynamics: the evolution is governed by the Mittag–Leffler function rather than an exponential, leading to subdiffusive or anomalous spreading laws for wave packets,

$\langle x^2 \rangle(t) \sim t^{2\alpha/\beta},$

where $\alpha$ (time) and $\beta$ (space) are the respective orders of the fractional derivatives (Gabrick et al., 2023). In two-level (Rabi-type) systems, this produces nonunitary relaxation with population spreading and power-law coherence decay.

4. Numerical and Experimental Approaches

4.1. High-Order Numerical Methods

Numerical solution of the FSE is challenged by the nonlocality of fractional operators. Recent advances include:

• Sixth-order split-step Fourier methods, supporting high-precision eigenfunction and tunneling rate computations for arbitrary $\alpha$ (Lewis et al., 2024).
• Spectral methods using Malmquist–Takenaka bases for unbounded domains, providing spectral or near-spectral convergence and efficient time-stepping (Strang/Yoshida splitting, ETDRK4) (Shen et al., 2022).
• Frozen Gaussian approximation (FGA) frameworks, retaining accuracy in highly oscillatory, semiclassical ($\varepsilon \ll 1$) regimes and handling singularities via regularized phase-space cutoffs (Chai et al., 2024).
• Petviashvili-type fixed-point iteration for stationary soliton solutions in the fractional nonlinear Schrödinger equation (fNLSE), combined with a spectral implementation of the fractional Laplacian (Bayindir et al., 2021).

4.2. Experimental Realizations

Experimental emulation of the FSE has been demonstrated in both spatial and temporal domains. In optics, a 4f-resonator with appropriately shaped aspherical phase masks realizes the space-fractional Schrödinger equation and permits direct excitation of fractional quantum harmonic oscillator (dual Airy) eigenmodes (Longhi, 2015).

Temporal-domain implementations employ femtosecond pulse shaping with programmable holograms to generate arbitrary input spectral phases and emulate propagation through "Lévy waveguides" with prescribed fractional group-velocity dispersion. Observed phenomena include solitary splitting, double Airy modes, and "rain-like" multi-pulse backgrounds, with robust "fractional-phase protection" for information encoding (Liu et al., 2022).

5. Nonlinear and Discrete Fractional Schrödinger Equations

The fractional nonlinear Schrödinger equation (fNLSE),

$$i\psi_t = -\tfrac{1}{2}(-\Delta)^{\alpha/2} \psi + V(x)\psi + \sigma|\psi|^2\psi,$$

supports families of stationary soliton solutions whose profiles and stability depend critically on $\alpha$. Lowering $\alpha$ concentrates energy, modifies modulational stability (Vakhitov–Kolokolov condition), and changes splitting and spreading behavior (Bayindir et al., 2021). Lattice models replace the nearest-neighbor hopping with fractional discrete Laplacians, producing long-range coupling, altered bandwidth, and nonmonotonic ballistic transport—quantified precisely via kernel decay and mean-square displacement analysis (Molina, 2019).

When singular potentials (e.g., delta or higher derivatives) are present, very weak solution theory ensures well-posedness and consistent weak limits through mollifier regularization. Notably, delta-like potentials induce particle accumulation, and stronger singularities can split wave packets, demonstrating new forms of quantum control and localization (Altybay et al., 2020).

6. Applications and Outlook

Fractional Schrödinger equations underpin modeling of quantum systems with anomalous transport and nonlocal interactions, relevant to multiscale/heterogeneous media, complex optical structures, and quantum information. Their signatures include power-law tails, enhanced tunneling, sub- and super-diffusion, and exceptional mode selectivity in engineered cavities. The mathematical structure also enables new approaches for simulation, signal processing, and the study of nonlinear, memory-laden quantum systems (Longhi, 2015, Liu et al., 2022, Iomin, 2021).

Rigorous theoretical frameworks, robust numerical solvers, and experimental hardware now support systematic exploration of these nonlocal quantum dynamics. Open challenges remain, particularly in the classification of nonlinear stationary states, dynamics under singular potentials, and the role of fractional operators in emergent quantum technologies and complex materials.