On the support of measures in multiplicative free convolution semigroups

In this paper, we study the supports of measures in multiplicative free semigroups on the positive real line and on the unit circle. We provide formulas for the density of the absolutely continuous parts of measures in these semigroups. The descriptions rely on the characterizations of the images of the upper half-plane and the unit disc under certain subordination functions. These subordination functions are $\eta$-transforms of infinitely divisible measures with respect to multiplicative free convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter

Rogue waves in a two-component Manakov system with variable coefficients and an external potential

We construct rogue waves (RWs) in a coupled two-mode system with the self-focusing nonlinearity of the Manakov type (equal SPM and XPM coefficients), spatially modulated coefficients, and a specially designed external potential. The system may be realized in nonlinear optics and Bose-Einstein condensates. By means of a similarity transformation, we establish a connection between solutions of the coupled Manakov system with spatially-variable coefficients and the basic Manakov model with constant coefficients. Exact solutions in the form of two-component Peregrine and dromion waves are obtained. The RW dynamics is analyzed for different choices of parameters in the underlying parameter space. Different classes of RW solutions are categorized by means of a naturally introduced control parameter which takes integer values.Comment: 9 pages, 5 figure

Ballistic Thermal Transistor of Dielectric Four-terminal Nanostructures

We report a theoretical model for a thermal transistor in dielectric four-terminal nanostructures based on mesoscopic ballistic phonon transport, in which a steady thermal flow condition of system is obtained to set up the temperature field effect of gate. In the environment, thermal flow shows the transisting behaviors at low temperatures: saturation, asymmetry, and rectification. The phenomena can be explained reasonably by the nonlinear variation of the temperature dependence of propagating phonon modes in terminals. The results suggest the possibility of the novel nano-thermal transistor fabrication

Deformed single ring theorems

Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $\Sigma=\Sigma_N$ such that $A\to a$ and $\Sigma\to \sigma$ in $\ast$-distribution for some operators $a$ and $\sigma$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $U\Sigma V^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $\sigma$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erd\H{o}s and Schnelli