Measuring the phonon-assisted spectral function by using a non-quilibrium three-terminal single-molecular device

The electron transport through a three-terminal single-molecular transistor (SMT) is theoretically studied. We find that the differential conductance of the third and weakly coupled terminal versus its voltage matches well with the spectral function versus the energy when certain conditions are met. Particularly, this excellent matching is maintained even for complicated structure of the phonon-assisted side peaks. Thus, this device offers an experimental approach to explore the shape of the phonon-assisted spectral function in detail. In addition we discuss the conditions of a perfect matching. The results show that at low temperatures the matching survives regardless of the bias and the energy levels of the SMT. However, at high temperatures, the matching is destroyed.Comment: 9 pages, 5 figure

Existence and multiplicity of positive solutions for a Schrodinger-Poisson system with a perturbation

In this paper we study the nonlinear Schrodinger-Poisson system with a perturbation: \begin{equation*} \begin{cases} -\Delta u+u+K( x) \phi u=\vert u\vert ^{p-2}u+\lambda f(x)\vert u\vert ^{q-2}u \text{in }\mathbb{R}^{3}, -\Delta \phi =K( x) u^{2} \text{in }\mathbb{R}^{3}, \end{cases} \end{equation*}% where $K$ and $f$ are nonnegative functions, $2\ge q\leq p\le 6$ and $p\ge 4$, and the parameter $\lambda \in \mathbb{R}$. Under some suitable assumptions on $K$ and $f$, the criteria of existence and multiplicity of positive solutions are established by means of the Lusternik-Schnirelmann category and minimax method

One-dimensional quantum channel in a graphene line defect

Using a tight-binding model, we study a line defect in graphene where a bulk energy gap is opened by sublattice symmetry breaking. It is found that sublattice symmetry breaking may induce many configurations that correspond to different band spectra. In particular, a gapless state is observed for a configuration which hold a mirror symmetry with respect to the line defect. We find that this gapless state originates from the line defect and is independent of the width of the graphene ribbon, the location of the line defect, and the potentials in the edges of the ribbon. In particular, the gapless state can be controlled by the gate voltage embedded below the line defect. Finally, this result is supported with conductance calculations. This study shows how a quantum channel could be constructed using a line defect, and how the quantum channel can be controlled by tuning the gate voltage embedded below the line defect.Comment: 8 pages, 10 figure

Normalized solutions for the Schr\"{o}dinger equation with combined Hartree type and power nonlinearities

We investigate normalized solutions for the Schr\"{o}dinger equation with combined Hartree type and power nonlinearities, namely \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u=\gamma (I_{\alpha }\ast \left\vert u\right\vert ^{p})|u|^{p-2}u+\mu |u|^{q-2}u & \quad \text{in}\quad \mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}}|u|^{2}dx=c, & \end{array}% \right. \end{equation*} where $N\geq 2$ and $c>0$ is a given real number. Under different assumptions on $\gamma ,\mu ,p$ and $q$, we prove several nonexistence, existence and multiplicity results. In particular, we are more interested in the cases when the competing effect of Hartree type and power nonlinearities happens, i.e. $\gamma \mu <0,$ including the cases $\gamma 0$ and Due to the different "strength" of two types of nonlinearities, we find some differences in results and in the geometry of the corresponding functionals between these two cases