# Leaky quantum graphs

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

**Funding:** Fully funded

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

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

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 R ^{3} with electrostatic and
Lorentz scalar delta-shell interactions*, Quantum Studies: Mathematics and
Foundations 6 (2019), 295-314.