arXiv subjects

Alexander Komech

Publications and source records attributed to Alexander Komech.

At least 37 records · Page 2Linked to original sources

arXiv2015

On linear stability of crystals in the Schroedinger-Poisson model

We consider the Schrödinger--Poisson--Newton equations for crystals with a cubic lattice and one ion per cell. We linearize this dynamics at the ground state and introduce a novel class of the ion charge densities which provide the stability of the linearized dynamics. This is the first result on linear stability for crystals. Our key result is the {\it energy positivity} for the Bloch generators of the linearized dynamics under a Wiener-type condition on the ion charge density. We also assume an additional condition which cancels the negative contribution caused by electrostatic instability. The proof of the energy positivity relies on a special factorization of the corresponding Hamilton functional. We show that the energy positivity can fail if the additional condition breaks down while the Wiener condition holds. The Bloch generators are nonselfadjoint (and even nonsymmetric) Hamilton operators. We diagonalize these generators using our theory of spectral resolution of the Hamilton ope\-rators {\it with positive definite energy} \ci{KK2014a,KK2014b}. Using this spectral resolution, we establish the stability of the linearized crystal dynamics.

math-ph↗

arXiv2015

Attractors of nonlinear Hamiltonian PDEs

This is a survey of results on long time behavior and attractors for nonlinear Hamiltonian partial differential equations, considering the global attraction to stationary states, stationary orbits, and solitons, the adiabatic effective dynamics of the solitons, and the asymptotic stability of the solitary manifolds. The corresponding numerical results and relations to quantum postulates are considered.

math-ph↗

arXiv2015

On the Hartree-Fock dynamics in wave-matrix picture

We introduce the Hamiltonian dynamics with the Hartree-Fock energy in new {\it wave-matrix} picture. Roughly speaking, the wave matrix is defined as the square root of the density matrix. The corresponding Hamiltonian equations are equivalent to an operator anticommutation equation. This wave-matrix picture essentially agrees with the density matrix formalism. Its main advantage is that it is Hamiltonian and allows an extension to infinite particle systems like crystals. Our main result is the existence of the global "reduced" wave-matrix dynamics for finite-particle molecular systems, and the energy and charge conservation laws. For the proof we extend known techniques, based on Hardy's and Sobolev's inequalitites, to the wave-matrix picture.

math-ph↗

arXiv2014

On Dynamical Justification of Quantum Scattering Cross Section

A~dynamical justification of quantum differential cross section in the context of long time transition to stationary regime for the Schrödinger equation is suggested. The problem has been stated by Reed and Simon. Our approach is based on spherical incident waves produced by a harmonic source and the long-range asymptotics for the corresponding spherical limiting amplitudes. The main results are as follows: i)~the convergence of spherical limiting amplitudes to the limit as the source increases to infinity, and ii) the universally recognized formula for the differential cross section corresponding to the limiting flux. The main technical ingredients are the Agmon--Jensen--Kato's analytical theory of the Green function, Ikebe's uniqueness theorem for the Lippmann--Schwinger equation, and some adjustments of classical asymptotics for the Coulomb potentials.

math-ph↗

arXiv2013

Weighted energy decay for magnetic Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of 3D Klein-Gordon equation with magnetic and scalar potentials. The decay extends the results obtained by Jensen and Kato for the Schroedinger equation with scalar potential. For the proof we develop the spectral theory of Agmon, Jensen and Kato and minimal escape velocities estimates of Hunziker, Sigal and Soffer.

math.AP↗

arXiv2013

On eigenfunction expansion of solutions to the Hamilton equations

We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert spaces with an indefinite metric. Our main result is an application to the eigenfunction expansion for the linearized relativistic Ginzburg-Landau equation.

math.AP↗

arXiv2013

On the Titchmarsh convolution theorem for distributions on the circle

We prove a version of the Titchmarsh convolution theorem for distributions on the circle. We show that the "naive form" of the Titchmarsh theorem could be violated, but that such a violation is only possible for the convolution of distributions which both possess certain symmetry properties.

math.FA↗

arXiv2012

On Lagrangian Theory for Rotating Charge Coupled to the Maxwell Field

We justify the Hamilton least action principle for the Maxwell-Lorentz equations with Abraham's rotating extended electron. The main novelty in the proof is application of the variational Poincare equations on the Lie group SO(3). The variational equations allow to derive the corresponding conservation laws from general Noether theory of invariants.

math-ph↗

arXiv2012

On wave theory of the photoeffect

The photoelectric effect has been discovered by H. Hertz, and studied experimentally by P. Lenard. First theoretical explanation has been done by A. Einstein who suggested the corpuscular theory of light introducing the "photons" which are particles of the light. In the framework of the Schrödinger theory the effect has been described first by G. Wentzel who calculated the angulardistribution of the photocurrent. The calculation relies on the perturbation procedure applied to the coupled Maxwell-Schrödinger equations.

quant-ph↗

arXiv2012

On nonlinear wave equations with parabolic potentials

We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations with a kink solutions. The potentials allow an exact description of the spectral properties for the linearized equation at the kink. This description is necessary for the study of the stability properties of the kinks.

math-ph↗

arXiv2012

Weighted decay for magnetic Schroedinger equation

We obtain a dispersive long-time decay in weighted norms for solutions of 3D Schroedinger equation with generic magnetic and scalar potentials. The decay extends the results obtained by Jensen and Kato for the Schroedinger equation without magentic potentials. For the proof we develop the spectral theory of Agmon, Jensen and Kato, extending the high energy decay of the resolvent to the magnetic Schroedinger equation.

math-ph↗

arXiv2012

Well-posedness, energy and charge conservation for nonlinear wave equations in discrete space-time

We consider the problem of discretization for the U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez \cite{MR0503140} conserves the positive-definite discrete analog of the energy if the grid ratio is $dt/dx\le 1/\sqrt{n}$, where $dt$ and $dx$ are the mesh sizes of the time and space variables and $n$ is the spatial dimension. We also show that if the grid ratio is $dt/dx=1/\sqrt{n}$, then there is the discrete analog of the charge which is conserved. We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use the energy conservation to obtain the a priori bounds for finite energy solutions, thus showing that the Strauss -- Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation with positive nonlinear term in the Hamiltonian is conditionally stable.

math.AP↗

arXiv2011

On convergence to equilibrium distribution for Dirac equation

We consider the Dirac equation in $\R^3$ with a potential, and study the distribution $μ_t$ of the random solution at time $t\in\R$. The initial measure $μ_0$ has zero mean, a translation-invariant covariance, and a finite mean charge density. We also assume that $μ_0$ satisfies a mixing condition of Rosenblatt- or Ibragimov-Linnik-type. The main result is the long time convergence of projection of $μ_t$ onto the continuous spectral space. The limiting measure is Gaussian.

math-ph↗

arXiv2010

On Scattering of Solitons for Maxwell Equation Coupled to a Particle

We establish long time soliton asymptotics for the nonlinear system of Maxwell equations coupled to a charged particle. The coupled system has a six dimensional manifold of soliton solutions. We show that in the long time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Maxwell equations. It is assumed that the charge density satisfies the Wiener condition. The proof further develops the general strategy based on the symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

math-ph↗

arXiv2010

Scattering of Solitons for Coupled Wave-Particle Equations

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

math-ph↗

arXiv2010

On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation

We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of a uniformly moving kink and dispersive part described by the free Klein-Gordon equation. The remainder decays in a global energy norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations.

math.AP↗

arXiv2010

On global attraction to quantum stationary states. Dirac equation with mean field interaction

We consider a U(1)-invariant nonlinear Dirac equation in dimension $n=3$, interacting with itself via the mean field mechanism. We analyze the long-time asymptotics of solutions and prove that, under certain generic assumptions, each finite charge solution converges as $t\to\pm\infty$ to the two-dimensional set of all "nonlinear eigenfunctions" of the form $ϕ(x)e\sp{-iωt}$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The research is inspired by Bohr's postulate on quantum transitions and Schrödinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schrödinger and Maxwell-Dirac equations.

math-ph↗

arXiv2010

On asymptotic stability of kink for relativistic Ginzburg-Landau equation

We prove the asymptotic stability of standing kink for the nonlinear relativistic wave equations of the Ginzburg-Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein-Gordon equation. The remainder converges to zero in a global norm. Crucial role in the proofs play our recent results on the weighted energy decay for the Klein-Gordon equations.

math.AP↗