# Vertex coupling effects in quantum graphs

**Advisor:** Pavel Exner (NPI CAS)

**Funding:** Basic scholarship + institutional supplement. Additional grant supplement might be specified later

**Website:** http://gemma.ujf.cas.cz/~exner/

**Contact:** exner@ujf.cas.cz

Quantum mechanics of particles whose motion is confined to a graphlike subset of the configuration space is an active field full of physical applications and interesting mathematical problems [1,2].

These models offer a lot of freedom in the choice of the way in which the wave functions can coupled at the graph vertices in order to preserve the probability current. Each of them can be given a physical meaning [3], however, most attention is traditionally paid to the simplest couplings satisfying the continuity condition. There are other cases of physical interest, for instance, an attempt to use graphs to model the anomalous Hall effect [4] motivated investigation of a coupling that violate manifestly the time-reversal invariance [5]. A potential PhD student can find in this area many other interesting open questions, e.g.

- it is not clear whether the topologically induced transport properties found in [5] can have a deeper reason than the mentioned noninvariance
- the coupling ‘texture’ of a graph can give rise to waveguide-type effects
- despite the universality of lattice graph spectra [6], there are examples with a finite number of gaps [2]; one asks how exceptional they are
- vertex coupling can have a profound influence on the quantum chaotic behavior of graph Hamiltonian spectra discovered in [7]
- for graph operators other than Schrödinger, such as Dirac, higher order or nonlinear, the coupling dependence is a largely unexplored area,

and many others. For inquiries you may use the e-mail address given above.

References:

[1] G. Berkolaiko, P. Kuchment: *Introduction to Quantum Graphs*,
Amer. Math. Soc., Provi- dence, R.I., 2013

[2] P. Exner: *Topologically induced spectral behavior: the example of
quantum graphs*, Proceedings of the ‘8th International Congress of Chinese
Mathematicians’, to appear’ arXiv:2003.06189

[3] P. Exner, O. Post: *A general approximation of quantum graph vertex
couplings by scaled Schrödinger operators on thin branched manifolds*,
Commun. Math. Phys. **322** (2013), 207–227

[4] P. Středa, J. Kučera: *Orbital momentum and topological phase
transformation*, Phys. Rev. **B92** (2015), 235152.

[5] P. Exner, M. Tater: *Quantum graphs with vertices of a preferred
orientation*, Phys. Lett. A382 (2018), 283–287

[6] R. Band, G. Berkolaiko: *Universality of the momentum band density of
periodic networks*, Phys. Rev. Lett. **113** (2013), 130404

[7] T. Kottos, U. Smilansky: *Quantum chaos on graphs*, Phys. Rev. Lett.
**79** (1997), 4794– 4797