Leaky quantum graphs

Quantum mechanics of particles confined to network-shaped regions – usually referred to as quantum waveguides or quantum graphs - attracted a lot of attention in the last three decades, both as a source of applications and of deep insigths into propertis of quantum theory [1-3].

Most of then the confinement in these models is "hard", so that quantum tuneling between different parts of the structure is neglected. A more realistic approach is to consider "leaky" structures in which we have a newtwork of potential "ditches", in the simplest case singular, delta-type ones. It is know that the geometry of the structure gives rise to an effective interation leading to spectral properties which defy our intuition based on macroscopic world experience [3, Chap. 10].

At the same time, may questions remain open, for instance

• properties of the spectrum in terms of the system geometry
• asymptotic expansions
• spectral optimization problems
• magnetic field effects
• relativistic dynamics of this type

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

References:

[1] J.T. Londergan, J.P. Carini, D.P. Murdock: Binding and Scattering in Two-Dimensional Systems. Applications to Quantum Wires, Waveguides and Photonic Crystals, Springer LNP m60, Berlin 1999
[2] G. Berkolaiko, P. Kuchment: Introduction to Quantum Graphs, Amer. Math. Soc., Provi- dence, R.I., 2013
[3] P. Exner, H. Kovařík: Quantum Waveguides, Springer International, Heidelberg 2015
[4] V. Lotoreichik, T. Ourmieres-Bonafos: On the bound states of Schrödinger operators with delta-interactions on conical surfaces, Comm. PDE 41 (2016), 999-1028.
[5] P. Exner, K. Pankrashkin: Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc, Comm. PDE 39 (2014), 193-212.
[6] B. Flamencourt, K. Pankrashkin: Strong coupling asymptotics for delta-interactions supported by curves with cusps, J. Math. Anal. Appl. 491} (2020), 124287.
[7] P. Exner, E.M. Harrell, M. Loss: Inequalities for means of chords, with application to isoperimetric problems, Lett. Math. Phys. 75 (2006), 242-233.
[8] P. Exner, S. Kondej: Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction, Lett. Math. Phys. 110 (2020), 735-751.
[9] D. Barseghyan, P. Exner: Magnetic field influence on the discrete spectrum of locally deformed leaky wires, Rep. Math. Phys. 88 (2021), 47-57.
[10] J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik: On Dirac operators in R³ with electrostatic and Lorentz scalar delta-shell interactions, Quantum Studies: Mathematics and Foundations 6 (2019), 295-314.