Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2506.22377 · Physica Scripta

Characteristic solutions of the chain of Vlasov equations

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Anton S. Medvedev

Published Jun 25, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schrödinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.

DOI
10.1088/1402-4896/ae82c3

Related papers

Dispersion chain of quantum mechanics equations

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

Abstract Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton–Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie–Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.

arXiv:2209.14069•Journal of Physics A Mathematical and Theoretical•Feb 20, 2023•7 citations

Equations with physical applications arising from the Vlasov chain

Paul Bracken

The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.

International Journal of Modern Physics B•Aug 20, 2020•0 citations

The properties of the first equation of the Vlasov chain of equations

E. E. Perepelkin, B I Sadovnikov, N G Inozemtseva

A derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme for the construction of the Schrödinger equation solution from the Vlasov equation solution and vice-versa is shown. A process of introduction of the potential to the Schrödinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the equation for movement of the medium within electromagnetic fields.

arXiv:1502.02062•Journal of Statistical Mechanics Theory and Experiment•May 18, 2015•29 citations

Exact time dependence of solutions to the time-dependent Schrödinger equation

Max A. Lohe

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

Journal of Physics A Mathematical and Theoretical•Dec 9, 2008•65 citations

Quantum propagator and characteristic equation in the presence of a chain of δ-potentials

A. Refaei, F. Kheirandish

The quantum propagator and characteristic equation in the presence of a chain of δ-potentials are obtained in the rectangular, cylindrical and spherical coordinate systems. The simplicity and efficiency of the method is illustrated via examples. As an application, the characteristic equation of a quantum harmonic oscillator confined to an infinite box is obtained. The roots of the characteristic equation, determining the energy eigenvalues of the restricted oscillator, are calculated approximately and compared with the existing numerical data. Due to the similarity between Schrödinger and diffusion equations, the problem can be considered as exact solutions of the diffusion equation in the presence of a chain of δ-potentials as boundary.

arXiv:1407.6170•International Journal of Modern Physics B•Apr 24, 2015•1 citation

The Wigner-Vlasov formalism for time-dependent quantum oscillator

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

Abstract This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space x , p , p ̇ , p ̈ .

arXiv:2305.06069•Physica Scripta•Sep 8, 2023•2 citations
Abstract A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schrödinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.
Abstract Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton–Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie–Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.
The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.
A derivation of the first Vlasov equation as a well-known Schrödinger equation for the probabilistic description of a system and families of the classic diffusion equations and heat conduction for the deterministic description of physical systems was inferred. A physical meaning of the phase of the wave function which is a scalar potential of the probabilistic flow velocity is demonstrated. Occurrence of the velocity potential vortex component leads to the Pauli equation for one of the spinar components. A scheme for the construction of the Schrödinger equation solution from the Vlasov equation solution and vice-versa is shown. A process of introduction of the potential to the Schrödinger equation and its interpretation are given. The analysis of the potential properties gives us the Maxwell equation, the equation of the kinematic point movement, and the equation for movement of the medium within electromagnetic fields.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
The quantum propagator and characteristic equation in the presence of a chain of δ-potentials are obtained in the rectangular, cylindrical and spherical coordinate systems. The simplicity and efficiency of the method is illustrated via examples. As an application, the characteristic equation of a quantum harmonic oscillator confined to an infinite box is obtained. The roots of the characteristic equation, determining the energy eigenvalues of the restricted oscillator, are calculated approximately and compared with the existing numerical data. Due to the similarity between Schrödinger and diffusion equations, the problem can be considered as exact solutions of the diffusion equation in the presence of a chain of δ-potentials as boundary.
Abstract This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space x , p , p ̇ , p ̈ .