Figure 1

(Color online) (a) The considered open TB system, which consists of a molecule with four sites (labeled by 1–4) connected to two leads (labeled by I and II) with serially connected sites (labeled by $i_{\eta }$ for lead $\eta$, where $i=0–\infty$ and $\eta =$ I and II). The hoppings are denoted by bonds. (b) The transmission probability $T$ at different energies for $V_2=0$ (solid line), where there is a LSC (located by a dotted line); and $V_2=0.05t$ (dash-dotted line) and $0.3t$ (dash-dot-dotted line), where there are no LSCs. Those system parameters not mentioned are referred to in the text.Reuse & Permissions

Figure 2

(Color online) (a) The considered open quantum graph, which consists of a 1D ring (formed by the branches labeled by II and III) connected to two leads (labeled by I and IV). A coordinate $x_{\eta }$ with positive direction indicated by the arrow is defined on branch $\eta$ ($\eta =$ I, II, III, and IV). (b) The transmission probability $T$ is plotted versus a dimensionless wave number $\kappa$ defined by $\kappa \equiv k(L_{\mathrm{II}}+L_{\mathrm{III}})∕\left(2\pi \right)$, for $\nu =0$, and $L_{\mathrm{II}}:L_{\mathrm{III}}=1:3$ (solid line) and $1:3.4$ (dash-dotted line). There is a LSC for the $1:3$ case (indicated by a dotted line), but not for the $1:3.4$ case in this energy range.Reuse & Permissions

Figure 3

(Color online) (a) The considered open 2D waveguide. Two leads (labeled by I and II) of width $W$ are attached at opposite sides of a square cavity (shaded region). The potential in the leads is kept at zero, and that in the cavity is kept at $V_{\mathrm{G}}$. A coordinate system $(x,y)$ is defined in the cavity as shown. (b) The transmission probability $T$ is plotted versus a dimensionless wave number $\kappa$ defined by $\kappa \equiv \sqrt{2mE}W∕(\pi \hslash )$, for the case shown in (a) (solid line) and the case with an additional $\delta$ potential $V(x,y)=v_0\delta (x-W∕6)\delta (y-W∕6)$ (dash-dotted line). In this energy range, LSCs (indicated by dotted lines) are found only in the case of no $\delta$ potential. The pairs of integers in the form $[m_x,m_y]$ indicate the profiles of the LSCs.Reuse & Permissions

Figure 4

The probability densities of two of the LSCs in Fig. 3b are shown in a section of the waveguide with the cavity at the center. The length of the section is three times the width. (a) The [2,2] LSC at $\kappa \simeq 1.74$ has sizable exponential tails in the leads. Correspondingly, it has also a considerable redshift from ${\kappa }_{\mathrm{SW}}(2,2)\simeq 2.23$, where ${\kappa }_{\mathrm{SW}}(m_x,m_y)$ is defined by ${\kappa }_{\mathrm{SW}}(m_x,m_y)\equiv \sqrt{2m{E}_{\mathrm{SW}}(m_x,m_y)}W∕(\pi \hslash )$. (b) The [1,3] LSC at $\kappa \simeq 2.51$ has smaller exponential tails in the leads, and also a lesser redshift from ${\kappa }_{\mathrm{SW}}(1,3)\simeq 2.64$.Reuse & Permissions