Fractional Schrodinger equation

Abstract

Properties of the fractional Schrodinger equation have been studied. We have proven the hermiticity of fractional Hamilton operator and established the parity conservation law for the fractional quantum mechanics. As physical applications of the fractional Schrodinger equation we have found the energy spectrum for a hydrogen-like atom - fractional ''Bohr atom'' and the energy spectrum of fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density has been developed and discussed. We also discuss the relationships between the fractional and the standard Schrodinger equations.

18 pages

References in corpus (2)

• Fractional Quantum Mechanics
• Collapse arrest and soliton stabilization in nonlocal nonlinear media

Cited by in corpus (255)

• Fractional Schrödinger equation in optics
• Time-space fabric underlying anomalous diffusion
• Time fractional Schrodinger equation
• Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$
• Fractional dynamics of systems with long-range interaction
• On the continuum limit for discrete NLS with long-range lattice interactions
• $\mathcal{P}\mathcal{T}$ symmetry in a fractional Schrödinger equation
• Fractional dynamics of coupled oscillators with long-range interaction
• Some properties of the one-dimensional Lévy crystal
• Nonlinear Fractional Dynamics on a Lattice with Long Range Interactions
• Diffraction-free beams in fractional Schrödinger equation
• Soliton dynamics in a fractional complex Ginzburg-Landau model
• One-dimensional solitons in fractional Schrödinger equation with a spatially modulated nonlinearity: nonlinear lattice
• Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian
• On distributions of functionals of anomalous diffusion paths
• Time Fractional Quantum Mechanics
• Dynamics with Low-Level Fractionality
• Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
• One-dimensional gap solitons in quintic and cubic-quintic Fractional nonlinear Schrödinger equations with a periodically modulated linear potential
• Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials
• Propagation dynamics of the circular Airy Gaussian vortex beams in the fractional nonlinear Schrödinger equation
• Fractional Choquard Equation with Critical Nonlinearities
• Prospecting Black Hole Thermodynamics with Fractional Quantum Mechanics
• Symmetry breaking of spatial Kerr solitons in fractional dimension
• Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• On fractional Schrodinger systems of Choquard type
• Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice
• On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
• Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities
• Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator
• Time Fractional Schrödinger Equation; Fox's H-functions and the Effective Potential
• Definition of the Riesz Derivative and its Application to Space Fractional Quantum Mechanics
• Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes
• Fractional Derivative as Fractional Power of Derivative
• Fractional Statistical Mechanics
• On the nature of the conformable derivative and its applications to physics
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation
• Chaotic and pseudochaotic attractors of perturbed fractional oscillator
• Magnetohydrodynamics of Fractal Media
• Nonholonomic Constraints with Fractional Derivatives
• Psi-Series Solution of Fractional Ginzburg-Landau Equation
• Collocation method for fractional quantum mechanics
• Quantum mechanics in fractional and other anomalous spacetimes
• Multiplicity and concentration results for a fractional Choquard equation via penalization method
• Quadratic fractional solitons
• Comment on "Fractional quantum mechanics" and "Fractional Schroedinger equation"
• The Two-Dimensional Fractional Discrete Nonlinear Schrodinger Equation
• The fractional symmetric rigid rotor
• Energy-dependent noncommutative quantum mechanics
• From Fractional Quantum Mechanics to Quantum Cosmology: An Overture
• Stable and oscillating solitons of $\mathcal{P}\mathcal{T}$-symmetric couplers with gain and loss in fractional dimension
• Inflation and fractional quantum cosmology
• Broadening quantum cosmology with a fractional whirl
• Comments on the Properties of Mittag-Leffler Function
• Path Integral Formulation for Lévy Flights - Evaluation of the Propagator for Free, Linear and Harmonic Potentials in the Over- and Underdamped Limits
• The higher order fractional Calderón problem for linear local operators: uniqueness
• q-deformed Lie algebras and fractional calculus
• Common aspects of q-deformed Lie algebras and fractional calculus
• Fractional Dynamics of Systems with Long-Range Space Interaction and Temporal Memory
• Levy flights and nonlocal quantum dynamics
• Spontaneous symmetry breaking and ghost states supported by the fractional nonlinear Schrödinger equation with focusing saturable nonlinearity and PT-symmetric potential
• Gauge Invariant Fractional Electromagnetic Fields
• Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrödinger equations
• Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative
• Fractional Almost Kahler - Lagrange Geometry
• Fractional dark energy
• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• Multiple solutions for a fractional $p$-Laplacian equation with sign-changing potential
• Fractional Holographic Dark Energy
• Remarks on the inhomogeneous fractional nonlinear Schrodinger equation
• Estimated Age of the Universe in Fractional Cosmology
• Bubbles and W-shaped solitons in Kerr media with fractional diffraction
• Conformal Anisotropic Mechanics
• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well
• Fractional Quantum Field Theory, Path Integral, and Stochastic Differential Equation for Strongly Interacting Many-Particle Systems
• Fractional random walk lattice dynamics
• Tunneling time in space fractional quantum mechanics
• New Integrable Multi-Lévy-Index and Mixed Fractional Nonlinear Soliton Hierarchies
• Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations
• Multiple Semiclassical Standing Waves for Fractional Nonlinear Schrödinger Equations
• Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary
• Anomalous Diffusion in One-Dimensional Disordered Systems: A Discrete Fractional Laplacian Method
• Time in quantum mechanics and the local non-conservation of the probability current
• Nonlinear fractional Schrödinger equations in one dimension
• Comment on: "On the consistency of solutions of the space fractional Schrödinger equation"
• Fedosov Quantization of Fractional Lagrange Spaces
• Second-harmonic generation in the system with fractional diffraction
• Zero mass case for a fractional Berestycki-Lions type problem
• Lévy flights in inhomogeneous environments
• Fractional Schrödinger equation and time dependent potentials
• New scattering features in non-Hermitian space fractional quantum mechanics
• Fractional Scalar Field Cosmology
• Quantum Cosmology of Fab Four John Theory with Conformable Fractional Derivative
• Green function of the double fractional Fokker-Planck equation: Path integral and stochastic differential equations
• Consistency Problem of the Solutions of the Space Fractional Schrödinger Equation
• Factorization of the Quantum Fractional Oscillator
• Localized modes in nonlinear fractional systems with deep lattices
• Fractional dynamic symmetries and the ground state properties of nuclei
• The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo
• Many-body forces with the envelope theory
• Modified Friedmann equations from fractional entropy
• Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling
• Repulsive Casimir Force from Fractional Neumann Boundary Conditions
• Emergence of fractal cosmic space from fractional quantum gravity
• Caputo Fractional Standard Map: Scaling Invariance Analyses
• Applications of density matrix in the fractional quantum mechanics
• A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions
• A fractional generalization of the classical lattice dynamics approach
• Electromagnetic coupling of strongly non-local quantum mechanics
• Complex Path Integrals and the Space of Theories
• Ground state solutions for a fractional Schrödinger equation with critical growth
• Global dynamics of small solutions to the modified fractional Korteweg-de Vries and nonlinear Schrödinger equations
• Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
• Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in ${\mathbb{R}}^N$
• Riemann-Liouville Fractional Einstein Field Equations
• Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls-Nabarro Barrier
• On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation
• Finite Temperature Casimir Effect for a Massless Fractional Klein-Gordon field with Fractional Neumann Conditions
• Conformal and Non-Minimal Couplings in Fractional Cosmology
• Infrared spectroscopy of diatomic molecules - a fractional calculus approach
• On the fractional NLS equation and the effects of the potential well's topology
• Multiplicity of solutions for fractional Schrödinger systems in ${\mathbb{R}}^N$
• Fractional Generalization of Kac Integral
• Self-accelerating beam dynamics in the space fractional Schrödinger equation
• On dyadic nonlocal Schrödinger equations with Besov initial data
• Gauge waves generation and detection in Aharonov-Bohm electrodynamics
• Solutions of the fractional Schrödinger equation with sign-changing nonlinearity
• Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
• On symmetrizing the ultraspherical spectral method for self-adjoint problems
• Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon
• Schrödinger-Maxwell equations driven by mixed local-nonlocal operators
• Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map
• Fractional discrete vortex solitons
• Testing Short Distance Anisotropy in Space
• An optimization problem for the first eigenvalue of the $p-$fractional laplacian
• Existence and multiplicity of solutions for a nonlinear Schrödinger equation with non-local regional diffusion
• Numerical study of anomalous diffusion of light in semi-crystalline polymer structures
• Aharonov-Bohm electrodynamics in material media: a scalar e.m. field cannot cause dissipation in a medium
• Are current discontinuities in molecular devices experimentally observable?
• Lévy-Schrödinger wave packets
• Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials
• Topological symmetry breaking of self--interacting fractional Klein--Gordon field on toroidal spacetime
• An existence result for singular fractional Kirchhoff-Schrödinger-Poisson system
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• The fractional Schrödinger equation with Hardy-type potentials and sign-changing nonlinearities
• Similarity solutions for a class of Fractional Reaction-Diffusion equation
• Some remarks on Relativistic Diffusion and the Spectral Dimension Criterion
• Existence results for fractional p-Laplacian problems via Morse theory
• Oscillating dipole with fractional quantum source in Aharonov-Bohm electrodynamics
• Fractional quantum mechanics meets quantum gravity phenomenology
• Fractional Quantum Heat Engine
• Multiplicity and concentration results for local and fractional NLS equations with critical growth
• Local time of Levy random walks: a path integral approach
• Non-homogeneous fractional Schrödinger equation
• Superfluid hydrodynamics in fractal dimension space
• Orbital stability of standing waves of a class of fractional Schrodinger equations with a general Hartree-type integrand
• One-dimensional L{é}vy Quasicrystal
• A Critique of Covariant Emergent Gravity
• Continuum limit for discrete NLS with memory effect
• Classical and quantum cosmology of Fab Four John theories
• High-frequency electromagnetic emission from non-local wavefunctions
• Fractionality and $\mathcal{P}\mathcal{T}$- symmetry in a square lattice
• Advanced manifold-metric pairs
• The second hyperpolarizability of systems described by the space-fractional Schrodinger equation
• Two-dimensional fractional discrete NLS equations: dispersion relations, rogue waves, fundamental and vortex solitons
• Existence and stability of standing waves for nonlinear Schrodinger systems involving the fractional Laplacian
• The weakly coupled fractional one-dimensional Schrödinger operator with index $1<\alpha \le 2$
• Fractional phase transition in medium size metal clusters and some remarks on magic numbers in gravitationally and weakly interacting clusters
• Factorization of the Riesz-Feller fractional quantum harmonic oscillators
• Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds
• Markov processes and generalized Schroedinger equations
• Non-Markovian Light-Matter Dynamics in the Time Fractional Jaynes-Cummings Model with Modulated Coupling
• On the discreet spectrum of fractional quantum hydrogen atom in two dimensions
• The Fractionary Schrödinger Equation, Green Functions and Ultradistributions
• Violation of f-sum Rule with Generalized Kinetic Energy
• Fractionality and PT-symmetry in an electrical transmission line
• Soft matter and fractional mathematics: insights into mesoscopic quantum and time-space structures
• Fractional dynamics in nonlinear magnetic metamaterials
• Asymptotic estimates of oscillatory integrals with general phase and singular amplitude: Applications to dispersive equations
• Existence of infinitely many solutions for the fractional Schrödinger- Maxwell equations
• Fractional entropy of the Brown-Kuchař dust in fractional anti-de Sitter quantum gravity
• Bound state solutions for the supercritical fractional Schrödinger equation
• Multiple solutions for a fractional Schrodinger equation with potentials
• Fractional derivative of composite functions: exact results and physical applications
• The scattering of fractional Schrödinger operators with short range potentials
• Fractional Schrödinger Equation with singular potentials of higher-order
• WKB expansion for a fractional Schrödinger equation with applications to controllability
• On a Fractional Schrödinger equation in the presence of Harmonic potential
• Interplay of fractionality and $\mathcal{P}\mathcal{T}$- symmetry on a 1D lattice
• Fractal fits to Riemann zeros
• Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
• Numerical Chaos in a Fractional Order Logistic Map
• KAM Theorem for a Hamiltonian system with Sublinear Growth Frequencies at Infinity
• Mesoscopic Fractional Quantum in Soft Matter
• Space-time fractional NLS equation
• Existence of infinitely many solutions for a class of fractional Schrödinger equations in ${\mathbb{R}}^N$ with combined nonlinearities
• Quantum uncertainty and energy flux in extended electrodynamics
• An efficient spectral method for the fractional Schrödinger equation on the real line
• Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential
• The fractional saturable impurity
• Characterizing and quantifying weak chaos in fractional dynamics
• Spectral and Dynamical Properties of the Fractional Nonlinear Schrödinger Equation under Harmonic Confinement
• Fractional nonlinear surface impurity in a 2D lattice
• A fractional Anderson model
• Principle action for particles faster than light
• Propagation dynamics of abruptly autofocusing circular Airy-Gaussian vortex beams in the fractional Schrödinger equation
• Entire sign-changing solutions to the fractional critical Schr{ö}dinger equation
• Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials
• Local well-posedness of semilinear space-time fractional Schrödinger equation
• Dispersive estimates for time and space fractional Schrödinger equations
• Ground states of some coupled nonlocal fractional dispersive PDEs
• Static hyperpolarizability of space-fractional quantum systems
• High-accuracy power series solutions with arbitrarily large radius of convergence for fractional nonlinear differential equations
• Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension
• Uniqueness of the fractional derivative definition
• A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation
• The fractional nonlinear electrical lattice
• The fractional nonlinear impurity: A Green function approach
• Quantum Cosmology in $f(Q)$ theory
• Airy-Gaussian vortex beams in the fractional nonlinear-Schrödinger medium
• A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations
• Remarks on the fractional inhomogeneous Hartree equation
• On some nonlinear fractional equations involving the Bessel operator
• Positive solutions for the fractional Schrödinger equations with logarithmic and critical nonlinearities
• Multiple positive solutions for the fractional Schrödinger-Poisson systems involving critical nonlinearities with potential
• Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity
• Normalized solutions to the fractional Kirchhoff equations with combined nonlinearities
• Electron Spin Precession for the Time Fractional Pauli Equation
• $L^p$ eigenfunction bounds for fractional Schrödinger operators on manifolds
• The fractional Schrödinger equation on compact manifolds: Global controllability results
• Symmetry and Monotonicity of Positive Solutions to Schrödinger Systems with Fractional $p$-Laplacian
• Stabilization of single- and multi-peak solitons in the fractional nonlinear Schroedinger equation with a trapping potential
• Global well-posedness of cubic fractional Schrödinger equations in one dimension
• The fractional d'Alembert's formulas
• Semi-classical analysis for Fractional Schrödinger Equations with fast decaying potenials
• Normalized solutions for a coupled fractional schrodinger system in low dimensions
• Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system in ${\mathbb{R}}^3$
• Upper bounds for critical coupling constants for binding some quantum many-body systems
• Blow up of fractional Schrödinger equations on manifolds with nonnegative Ricci curvature
• Covariance in Fractional Calculus
• Generalised fractional Rabi problem
• Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels
• Fractional uncertainty
• Constraining fractionality using some observational tests
• Asymptotically linear fractional Schrodinger equations
• Normalized ground state solutions for the fractional Sobolev critical NLSE with an extra mass supercritical nonlinearity
• On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension
• Higher topological type semiclassical states for fractional nonlinear elliptic equations
• Investigation of Dephasing In an Open Quantum System under Chaotic Influence via a Fractional Kohn-Sham Scheme
• Soliton dynamics for fractional Schrodinger equations
• Propagation dynamics of radially polarized symmetric Airy beams in the fractional Schrödinger equation
• Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent
• Concentrating phenomenon for fractional nonlinear Schrödinger-Poisson system with critical nonlinearity