Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal attractor for 1D Dirac field coupled to nonlinear oscillator
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The long-time asymptotics is analyzed for all finite energy solutions to a model $mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {it each finite energy solution} converges as $ttopminfty$ to the set of all `nonlinear eigenfunctions of the form $(psi_1(x)e^{-iomega_1 t},psi_2(x)e^{-iomega_2 t})$. The {it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on emph{inflation of spectrum by the nonlinearity}. We show that any {it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $omega_jin[-m,m]$, $j=1,2$.
rate research
Read More
In this paper we present the tanh method to obtain exact solutions to coupled MkDV system. This method may be applied to a variety of coupled systems of nonlinear ordinary and partial differential equations.
The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrodinger equation. This method straightforwardly yields the correct Schrodinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.
Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli-Dirac one and, in contrast to the discrete case, critical energies are also found for the continuous Dirac case with positive mass.
This paper is a natural continuation of the previous paper cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=alpha/x^2$, $alphageq-1/4$, were constructed. In this paper, we present generalized oscillator representations for all generalized Calogero Hamiltonians with potential $V(x)=g_{1}/x^2+g_{2}x^2$, $g_{1}geq-1/4$, $g_{2}>0$. These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian, representation that explicitly determines the ground state and the ground-state energy. For generalized Calogero Hamiltonians with coupling constants $g_1<-1/4$ or $g_2<0$, generalized oscillator representations do not exist in agreement with the fact that the respective Hamiltonians are not bounded from below.
It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translational invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analiticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the abscence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Arakis result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Arakis result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
