Fractional-time Schrödinger equation: Fractional dynamics on a comb

  title={Fractional-time Schr{\"o}dinger equation: Fractional dynamics on a comb},
  author={Alexander Iomin},
  journal={Chaos Solitons \& Fractals},
  • A. Iomin
  • Published 1 May 2011
  • Mathematics, Physics
  • Chaos Solitons & Fractals
The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Schrodinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion ∂∂t→∂α∂tα can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for α = 1/2, the FTSE is a particular case of the quantum comb… 
Space-Time Fractional Schrödinger Equation With Composite Time Fractional Derivative
Abstract The fractional Schrödinger equation has recently received substantial attention. We generalize the fractional Schrödinger equation to the Hilfer time derivative and the Caputo space
Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors
Abstract.A new fractional Schrödinger equation is constructed from path integral based on the notions of fractional velocity recently introduced in literature and the concept of fractional actionlike
Time Fractional Schrodinger Equation Revisited
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle”
Time fractional Schrödinger equation: Fox's H-functions and the effective potential
After introducing the formalism of the general space and time fractional Schrodinger equation, we concentrate on the time fractional Schrodinger equation and present new results via the elegant
The time-dependent Schrödinger equation in three dimensions under geometric constraints
We consider a quantum motion governed by the time-dependent Schrodinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrodinger equations for the reduced
The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion
Abstract We propose a theoretical model, based on a generalized Schrodinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The
Fractional evolution in quantum mechanics
  • A. Iomin
  • Physics
    Chaos, Solitons & Fractals: X
  • 2019
Abstract Fractional time (non-unitary) quantum mechanics is discussed for both Schrodinger and Heisenberg representations of quantum mechanics. A correct form of the fractional Schrodinger equation
The transition of energy and bound states in the continuum of fractional Schrödinger equation in gravitational field and the effect of the minimal length
It is proved the existence of bound states in the continuum (BICs) for the fractional quantum system in the Earth’s gravitational field and compare the authors' BICs with those previous ones and provide the energy characteristic of small mass particles.
Generalized time-dependent Schrödinger equation in two dimensions under constraints
We investigate a generalized two-dimensional time-dependent Schrodinger equation on a comb with a memory kernel. A Dirac delta term is introduced in the Schrodinger equation so that the quantum
Initial Value Problem for a Caputo Space-time Fractional Schrodinger Equation for the Delta Potential
In this paper, we investigate the initial value problem for a Caputo spacetime fractional Schrödinger equation for the delta potential. To solve this equation, we use the joint Laplace transform on


Fractional-time quantum dynamics.
  • A. Iomin
  • Mathematics, Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2009
It is shown that for alpha=1/2 the fractional SE is isospectral to a comb model and an analytical expression for the Green's functions of the systems are obtained.
Space-time fractional Schrödinger equation with time-independent potentials
Abstract We develop a space–time fractional Schrodinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrodinger equation in this
Time fractional Schrödinger equation
The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to
Generalized fractional Schrödinger equation with space-time fractional derivatives
In this paper the generalized fractional Schrodinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by
Fractals and quantum mechanics.
The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics.
Fractional Schrodinger wave equation and fractional uncertainty principle
Free particle wavefunction of the fractional Schrödinger wave equation is obtained. The wavefunction of the equation is represented in terms of generalized three-dimensional Green’s function that
The random walk's guide to anomalous diffusion: a fractional dynamics approach
Abstract Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems
Superdiffusion on a comb structure.
It is shown that an inhomogeneous convection flow is a mechanism for the realization of the Lévy-like process, which leads to superdiffusion of particles on the comb structure, an enhanced one with an arbitrary large transport exponent, but all moments are finite.
Anomalous diffusion and drift in a comb model of percolation clusters
The example of a comb structure is used in a study of the influence of "dead ends" on the diffusion and drift of particles along percolation clusters. It is shown that the equation for the
Explicit time-dependent Schrodinger propagators
The authors compute explicitly the time-dependent Schrodinger and heat propagator for the potentials -( lambda 2m(m+1))/cosh2 lambda x, a delta function potential, several cases of periodic delta