--- title: Time-Fractional Schrödinger Equation url: https://www.emergentmind.com/topics/time-fractional-schrodinger-equation-tfse type: topic --- # Time-Fractional Schrödinger Equation 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 [2504.10026], [1108.6178], [1103.3295], [1907.04227]. 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 [1108.6178]. 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 \(0<\alpha<1\) and \(D_\alpha\to \hbar/(2m)\) as \(\alpha\to 1\) [1103.3295]. A broader space-time fractional framework couples a Caputo time derivative to the quantum Riesz-Feller space-fractional derivative. For time-independent potentials, \[ C_{\alpha}D_x^{\alpha,\theta}\Psi(x,t)+V(x)\Psi(x,t)=(i\hbar)^{\beta}\,{}^{C}D_t^{\beta}\Psi(x,t), \] with \(0<\alpha\le 2\), \(0<\beta\le 1\), and \(|\theta|\le \min\{\alpha,2-\alpha\}\). Separation of variables gives the pure time-fractional equation \[ (i\hbar)^{\beta}\,{}^{C}D_t^{\beta}T(t)=E\,T(t), \] together with a stationary space-fractional eigenproblem [1709.06198]. Other derivative choices materially alter the theory. The Hadamard–Caputo derivative is used in the Cauchy problem \[ i^\alpha {}^{HC}D_{a+}^{\alpha} u(t,x)+\Delta u(t,x)=\lambda(\ln(t/a))^\gamma |u(t,x)|^p, \] where the logarithmic kernel is adapted to multiplicative time scaling [2201.11229]. A conformable variant replaces the nonlocal Caputo operator by \[ {}_a^{\mathrm{Co}}D_t^\beta f(t)=t^{1-\beta}\frac{df(t)}{dt}, \] leading to equations of the form \[ i\hbar_\beta\,{}_0^{\mathrm{Co}}D_t^\beta |\psi(x,t)\rangle=H_\beta |\psi(x,t)\rangle \] in recent open-system studies [2312.10488]. By contrast, a Riemann–Liouville-type derivative with lower bound \(-\infty\) has been used to define \[ D_t^\alpha u(t)=(iA)^\alpha u(t), \] which is shown to be equivalent to the first-order equation \[ i\,\partial_t u(t)=A^{1/\alpha}u(t) \] for a positive self-adjoint operator \(A\) [1805.06844]. ## 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 \[ {}^{C}D_t^\alpha T(t)=i^\alpha A_n T(t) \] has solution \[ T(t)=E_\alpha\!\left(i^\alpha A_n t^\alpha\right),\qquad E_\alpha(z)=\sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k+1)}, \] and derives this both by Laplace transform and by Bromwich inversion with Fox \(H\)-functions [1103.3295]. In the Hilbert-space setting, if \(A\) is positive self-adjoint and \(u_0\in D(A)\), the strong solution of \[ {}^{C}D_t^\alpha u(t)=(-i)^\alpha A u(t),\qquad u(0)=u_0, \] is \[ u(t)=U_\alpha(t)u_0,\qquad U_\alpha(t)=E_\alpha\!\big(( -it)^\alpha A\big), \] constructed by the spectral theorem and functional calculus [1611.08899]. In Fourier variables, the free-particle solution on \(\mathbb R^n\) takes the form \[ \widehat{u}(t,\xi)=E_\alpha\!\big(( -it)^\alpha |\xi|^2\big)\widehat{u}_0(\xi), \] so the linear TFSE is governed by a Mittag–Leffler Fourier multiplier rather than the standard \(e^{-it|\xi|^2}\) phase [1611.08899]. For \(1<\alpha<2\), the higher-order equation \[ i\,{}^{C}D_t^\alpha u(t,x)=-\Delta u(t,x)+f(t,x) \] is represented by oscillatory kernels with symbols \[ S_\alpha(t,\xi)=E_{\alpha,1}\big(i^{-\alpha}t^\alpha |\xi|^2\big),\quad Q_\alpha(t,\xi)=t\,E_{\alpha,2}\big(i^{-\alpha}t^\alpha |\xi|^2\big),\quad P_\alpha(t,\xi)=i^{-\alpha}t^{\alpha-1}E_{\alpha,\alpha}\big(i^{-\alpha}t^\alpha |\xi|^2\big), \] which are used to derive Hölder regularity and pointwise convergence to initial data [1907.04227]. 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 \(H\)-functions, while also recovering the Mittag–Leffler time factor for the separated temporal ODE [1712.10079]. Dong constructs a fractional Green’s function for the time-dependent scattering problem, expresses it in Fox \(H\)-function form and in a computable series form, and derives an asymptotic scattered wave with correction of every order [1301.3206]. ## 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 \[ i h\,\partial_t^\alpha \psi=\hat H\psi, \] 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 [1108.6178]. In the exact quantum comb model, the \(\alpha=1/2\) 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 \(\partial_t\mapsto \partial_t^\alpha\) can discard essential dynamical information [1108.6178]. Bayın’s analysis makes the non-unitary content explicit. For a normalized separable solution, \[ P(t)=\int |\psi(x,t)|^2\,dx=|T(t)|^2, \] and the Mittag–Leffler time factor yields \[ P(t)\approx 1+\frac{2\cos(\pi\alpha/2)}{\Gamma(1+\alpha)}A_n t^\alpha + O(t^{2\alpha}) \] for short times and \[ P(t)\sim c\,t^{-2\alpha} \] for long times, so the total probability is less than one and decays with time when \(A_n<0\) [1103.3295]. 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 \(\beta\neq 1\) [1703.00301]. Two distinct responses to this difficulty appear in the literature. One is structural reformulation: the Riemann–Liouville-type equation \[ D_t^\alpha u(t)=(iA)^\alpha u(t) \] with lower terminal \(-\infty\) is shown to be equivalent to \[ i\,\partial_t u(t)=A^{1/\alpha}u(t), \] so the evolution operator \[ U(t)=\exp\big(i t A^{1/\alpha}\big) \] forms a strongly continuous unitary group and conserves the \(L^2\) norm [1805.06844]. 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 [2208.13858]. ## 4. Nonlinear TFSEs, regularity theory, and nonexistence results Nonlinear TFSEs have been studied in several analytically distinct forms. For a Hartree perturbation on \(\mathbb R^n\), \[ i\,D_t^\alpha u(t,x)=(-\Delta)^{\beta/2}u(t,x)+\lambda\,J^{1-\alpha}\Big((K_\gamma * |u|^2)(x)\,u(t,x)\Big),\qquad u(0,x)=u_0(x), \] with \(n\ge 2\), \(\alpha\in(0,1)\), \(\gamma\in(0,n)\), \(\gamma/2\le \beta<1\), and \(u_0\in H^\beta(\mathbb R^n)\), local well-posedness is established in \(C([0,T];H^\beta(\mathbb R^n))\). The mild solution is \[ u(t)=E_\alpha\big(-i\,t^\alpha(-\Delta)^{\beta/2}\big)u_0 +\lambda\int_0^t E_\alpha\big(-i\,(t-s)^\alpha(-\Delta)^{\beta/2}\big)\Big((K_\gamma*|u(s)|^2)u(s)\Big)\,ds, \] and the data-to-solution map is continuous [1907.03021]. The existence theory is complemented by sharp nonexistence results in other fractional-time settings. For the Hadamard–Caputo equation \[ i^\alpha {}^{HC}D_{a+}^{\alpha} u(t,x)+\Delta u(t,x)=\lambda(\ln(t/a))^\gamma |u(t,x)|^p,\qquad u(a,x)=f(x)\in L^1(\mathbb R^N;\mathbb C), \] Alotaibi, Jleli, Ragusa, and Samet prove that no global weak solution exists under explicit sign conditions on the initial data. One sufficient criterion is \[ \gamma>-\alpha,\qquad \gamma(N\alpha-2)<2\alpha,\qquad \max\Bigl\{1,1+\frac{\gamma}{\alpha}\Bigr\}\alpha\). 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 [2401.07117]. 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 \(H\)-functions, spectral functional calculus, test-function methods, or history-dependent discretizations—are appropriate for its study. Source: https://www.emergentmind.com/topics/time-fractional-schrodinger-equation-tfse