**Author details**

Jan Smotlacha1,2\* and Richard Pincak3,1

\*Address all correspondence to: smota@centrum.cz

1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, Russia

2 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Breho‐ va, Prague, Czech Republic

3 Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republic

### **References**

In **Figure 21**, LDOS of zero modes is shown for a varying distance from the wormhole bridge in the units of the radius *a* of the wormhole center. It was also acquired in the numerical way. For the unperturbed case (0 defects), the resulting plot resembles a line. In reference [27], the exponential solution is found for this case but with a very slow increase, so this could be that case. It is also seen from the plot that for the increasing number of the defects, the solution is approaching expressions in Eqs. (54) and (55)forthe zero modes ofthe unperturbed wormhole. Of course, the massive fermions could also appear in the case of the perturbed wormhole. We will not perform a detailed derivation of the electronic structure for the case of this eventual‐ ity, and we only note that the corrections to LDOS would be an analogy of the corrections

We performed the calculations of the electronic structure for the graphitic nanocone and the graphene wormhole. In the first case, our aim was to find the quadratically integrable solution that includes the boundary effects and considers the real geometry. This goal was partially achieved, but we need to verify the properties of the found solution close to the tip. The precision of the calculations could be improved by the better choice of the corresponding geometry, by the consideration of the discretion of the energetic spectrum coming from the finite size of the nanostructure, and by the inclusion of next effects coming from the overlap of the neighboring atomic orbitals close to the tip [2]. The localization of the electrons shown in **Figures 11** and **12**, especially in the case of three defects, makes the graphitic nanocone a possible candidate for the construction of the scanning probe in atomic force microscopy.

In the second case of the graphene wormhole, we presented the mathematical motivation for our prediction of the effects that should appear close to the wormhole bridge. Our predic‐ tions will be verified with the help of the geometric optimizations and ab initio calculations.

1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna,

2 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Breho‐

3 Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republic

On this base, the most suitable candidates for the experiments will be chosen.

shown in **Figure 17**.

54 Recent Advances in Graphene Research

**5. Conclusion**

**Author details**

Moscow region, Russia

va, Prague, Czech Republic

Jan Smotlacha1,2\* and Richard Pincak3,1

\*Address all correspondence to: smota@centrum.cz

