A family of anisotropic integral operators and behaviour of its maximal eigenvalue

We study the family of compact integral operators $\mathbf K_\beta$ in $L^2(\mathbb R)$ with the kernel K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)}, depending on the parameter $\beta >0$, where $\Theta(x, y)$ is a symmetric non-negative homogeneous function of degree $\gamma\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_\beta$ of $\mathbf K_\beta$: M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}), \beta\to 0, where $\lambda_1$ is the lowest eigenvalue of the operator $\mathbf A = |d/dx| + \Theta(x, x)/2$. A central role in the proof is played by the fact that $\mathbf K_\beta, \beta>0,$ is positivity improving. The case $\Theta(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.Comment: 16 page

Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$

Noncommutativity and theta-locality

In this paper, we introduce the condition of theta-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutator of observables behaves at large spacelike separation like $\exp(-|x-y|^2/\theta)$, where $\theta$ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely corresponds to the Moyal star product and prove that this space is a topological algebra under the star product. As an example, we consider the simplest normal ordered monomial $:\phi\star\phi:$ and show that it obeys the theta-locality condition.Comment: LaTeX, 17 pages, no figures; minor changes to agree with published versio

Chaotic Spin Dynamics of a Long Nanomagnet Driven by a Current

We study the spin dynamics of a long nanomagnet driven by an electrical current. In the case of only DC current, the spin dynamics has a sophisticated bifurcation diagram of attractors. One type of attractors is a weak chaos. On the other hand, in the case of only AC current, the spin dynamics has a rather simple bifurcation diagram of attractors. That is, for small Gilbert damping, when the AC current is below a critical value, the attractor is a limit cycle; above the critical value, the attractor is chaotic (turbulent). For normal Gilbert damping, the attractor is always a limit cycle in the physically interesting range of the AC current. We also developed a Melnikov integral theory for a theoretical prediction on the occurrence of chaos. Our Melnikov prediction seems performing quite well in the DC case. In the AC case, our Melnikov prediction seems predicting transient chaos. The sustained chaotic attractor seems to have extra support from parametric resonance leading to a turbulent state

Whitney’s extension theorem for ultradifferentiable functions of Beurling type

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43941/1/11512_2006_Article_BF02386123.pd

Twisted convolution and Moyal star product of generalized functions

We consider nuclear function spaces on which the Weyl-Heisenberg group acts continuously and study the basic properties of the twisted convolution product of the functions with the dual space elements. The final theorem characterizes the corresponding algebra of convolution multipliers and shows that it contains all sufficiently rapidly decreasing functionals in the dual space. Consequently, we obtain a general description of the Moyal multiplier algebra of the Fourier-transformed space. The results extend the Weyl symbol calculus beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure

Helicity sensitive terahertz radiation detection by field effect transistors

Terahertz light helicity sensitive photoresponse in GaAs/AlGaAs high electron mobility transistors. The helicity dependent detection mechanism is interpreted as an interference of plasma oscillations in the channel of the field-effect-transistors (generalized Dyakonov-Shur model). The observed helicity dependent photoresponse is by several orders of magnitude higher than any earlier reported one. Also linear polarization sensitive photoresponse was registered by the same transistors. The results provide the basis for a new sensitive, all-electric, room-temperature and fast (better than 1 ns) characterisation of all polarization parameters (Stokes parameters) of terahertz radiation. It paves the way towards terahertz ellipsometry and polarization sensitive imaging based on plasma effects in field-effect-transistors.Comment: 7 pages, 4 figure

Sufficient conditions for the convergence of the Magnus expansion

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation $Y' = A(t) Y$. The first one provides a bound on the convergence domain based on the norm of the operator $A(t)$. The second condition links the convergence of the expansion with the structure of the spectrum of $Y(t)$, thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.Comment: 20 page

Wavefronts may move upstream in doped semiconductor superlattices

In weakly coupled, current biased, doped semiconductor superlattices, domain walls may move upstream against the flow of electrons. For appropriate doping values, a domain wall separating two electric field domains moves downstream below a first critical current, it remains stationary between this value and a second critical current, and it moves upstream above. These conclusions are reached by using a comparison principle to analyze a discrete drift-diffusion model, and validated by numerical simulations. Possible experimental realizations are suggested.Comment: 12 pages, 11 figures, 2-column RevTex, Phys. Rev. E 61, 1 May 200