arXiv subjects
Publications and source records attributed to A. I. Komech.
We consider dissipation operators used in Quantum Optics for the description of quantum spontaneous emission in the context of damped driven Jaynes-Cummings equations. The equations describe quantised one-mode Maxwell field coupled to a two-level molecule. Our main result is the symmetry and nonpositivity of basic dissipation operator of Quantum Optics.
We consider damped driven Maxwell-Bloch equations for a single-mode Maxwell field coupled to a two-level molecule. The equations are used for semiclassical description of the laser action. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of averaged reduced equations in the interaction picture. We calculate all harmonic states and analyse their stability. Our calculations rely on the Hopf reduction by the gauge symmetry group U(1). The asymptotics follow by an extension of the averaging theory of Bogolyubov--Eckhaus--Sanchez-Palencia onto dynamical systems on manifolds.The key role in the application of the averaging theory is played by a special a priori estimate.
The article addresses the damped driven Jaynes-Cummings for quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators. Our main result is the construction of a contraction dynamical semigroup in the Hilbert space of Hermitian Hilbert-Schmidt operators in the case of a nonpositive dissipation operator and time-independent pumping. All trajectories of the semigroup are generalised solutions to the Jaynes-Cummings equations. As a key example, we prove nonpositivity for the basic dissipation operator of Quantum Optics.
The article concerns damped driven Jaynes-Cummings equation which describes quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators, and their structures correspond to the theory of completely positive and trace preserving generators (CPTP) of Lindblad and Kossakowski & al. Our main result is the construction of global generalised solutions with values in the Hilbert space of nonnegative Hermitian Hilbert-Schmidt operators in the case of time-dependent pumping. The proofs rely on finite-dimensional approximations of the annihilation and creation operators.
We consider the Maxwell field coupled to a single rotating charge. This Hamiltonian system admits soliton-type solutions, where the field is static, while the charge rotates with constant angular velocity. We prove that any solution of finite energy converges, in suitable local energy seminorms, to the corresponding soliton in the long time limit.
A space-periodic ground state is shown to exist for lattices of point ions in $\R^3$ coupled to the Schrödinger and scalar fields. The coupling requires the renormalization of the selfaction because of the singularity of the Coulomb potential. The ground state is constructed by minimization of the renormalized energy per cell. This energy is bounded from below when the charge of each ion is positive. The elementary cell is necessarily neutral.
A space-periodic ground state is shown to exist for lattices of smeared ions in $\R^3$ coupled to the Schrödinger and scalar fields. The elementary cell is necessarily neutral. The 1D, 2D and 3D lattices in $\R^3$ are considered, and a ground state is constructed by minimizing the energy per cell. The case of a 3D lattice is rather standard, because the elementary cell is compact, and the spectrum of the Laplacian is discrete. In the cases of 1D and 2D lattices, the energy functional is differentiable only on a dense set of variations, due to the presence of the continuous spectrum of the Laplacian that causes the infrared divergence of the Coulomb bond. Respectively, the construction of electrostatic potential and the derivation of the Schrödinger equation for the minimizer in these cases require an extra argument. The space-periodic ground states for 1D and 2D lattices give the model of the nanostructures similar to the carbon nanotubes and graphene respectively.
We consider the scattering of an arbitary incident plane wave which is a tempered distribution by a two-dimensional wedge.
We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points.
The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator; mathematically the system under study is a nonlinear Schrödinger equation, whose nonlinear term includes a Dirac delta. The coupled system is invariant with respect to the phase rotation group U(1). This article, which extends the results of a previous one, provides a proof of asymptotic stability of solitary wave solutions in the case that the linearization contains a single discrete oscillatory mode satisfying a non-degeneracy assumption of the type known as the Fermi Golden Rule.
We consider the dynamics of a field coupled to a harmonic crystal with $n$ components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. Moreover, initial correlation functions are translation-invariant with respect to the discrete subgroup $\Z^d$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$, where $μ_\infty$ is translation-invariant with respect to the subgroup $\Z^d$.
We consider the Dirac equation in $\R^3$ with constant coefficients and study the distribution $μ_t$ of the random solution at time $t\in\R$. It is assumed that the initial measure $μ_0$ has zero mean, a translation-invariant covariance, and 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 convergence of $μ_t$ to a Gaussian measure as $t\to\infty$. The proof uses the study of long time asymptotics of the solution and S.N. Bernstein's ``room-corridor'' method.
Solitary waves of relativistic invariant nonlinear wave equation with symmetry group U(1) are considered. We prove that the energy-momentum relation for spherically symmetric solitary waves coincides with the Einstein energy-momentum relation for point particles.
Consider the wave equation with constant or variable coefficients in $\R^3$. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_3\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the random solution at a time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means central limit theorem for the wave equation. The proof is based on the Bernstein `room-corridor' argument. The application to the case of the Gibbs measures $μ_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density formally is $-\infty\cdot (0,0,T_+ -T_-)$ for the Gibbs measures, and it is finite and equals to $-C(0,0,T_+ -T_-)$ with $C>0$ for the convolution with a nontrivial test function.
Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $μ_t$ of the random solution at time $t\in\R$. We assume that the initial probability measure $μ_0$ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that $μ_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using an `averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.
The paper considers the wave equation, with constant or variable coefficients in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution $μ_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed that the initial measure $μ_0$ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that $μ_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure $μ_\infty$ as $t\to\infty$, which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.
We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components,$d,n \ge 1$. The initial date is a random function with finite mean density of the energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_d\to\pm\infty$, with the distributions $μ_\pm$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian translation-invariant measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green function and on Bernstein's `room-corridor' argument. The application to the case of the Gibbs measures $μ_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density is $- (0,...,0,C(T_+ - T_-))$ with some positive constant $C>0$ what corresponds to Second Law.
We consider the dynamics of a harmonic crystal in $d$ dimensions with $n$ components, $d,n$ arbitrary, $d,n\ge 1$, and study the distribution $μ_t$ of the solution at time $t\in\R$. The initial measure $μ_0$ has a translation-invariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt- resp. Ibragimov-Linnik type mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$. The proof is based on the long time asymptotics of the Green's function and on Bernstein's ``room-corridors'' method.