Mathematical Models of Metamaterials

Equations of quantum mechanics involve numerous effects of a potential use, some of them still unexplored. This applies, in particular, to relations between geometry and topology of systems on the one hand and their spectral and transport properties on the other, with applications to construction of artificial materials with prescribed properties. The proposed project aims at mathematical investigation of several classes of models of this type; a prospective PhD student can find in this area many interesting open questions, for instance:

• Quantum waveguides, both ‘hard’ [1] and ‘soft’ [2], exhibit geometrically induced localized states; one may ask what kinds of motion one would observe in networks of various structure composed of them.
• Quantum graphs [3], when arranged into lattices, have transport properties depending on the way in which the wave functions in the vertices of the graph are matched. In this way, one can produce, e.g., guided states along a perturbation [4] or to model a material with a negative refractive index through an interface of two lattices [5].
• Another challenge is to construct a quantum graph model of anomalous Hall effect which, in contrast to [6], would contain wave functions matching violating the time reversal invariance, allowing thus a justification from the first principles – and this is by far not all! On the technical side, what is required is the knowledge of quantum mechanics and basics of operator theory.

Inquiries are welcome; you may use the e-mail address indicated above.

References:

[1] P. Exner, H. Kovařík: Quantum Waveguides, Springer International, Heidelberg 2015
[2] P. Exner, S. Vugalter: Bound states in bent soft waveguides, J. Spectral Theory 14 (2024), 427–457
[3] G. Berkolaiko, P. Kuchment: Introduction to Quantum Graphs, Amer. Math. Soc., Providence, R.I., 2013
[4] M. Baradaran, P. Exner, A. Khrabustovskyi: Quantum lattice transport along an infinitely extended perturbation, J. Phys. A.: Math. Theor. 58 (2025), 145203 (27pp)
[5] T. Lawrie, G. Tanner, D. Chronopoulos: A quantum graph approach to metamaterial design, Sci. Rep. 12 (2022), 18005
[6] P. Středa, K. Výborný: Anomalous Hall conductivity and quantum friction, Phys. Rev. B107 (2023), 014425