# Soft quantum waveguides

**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 confined to regions of a fixed size in one
direction and extended in the others – for brevity we usually speak about
*quantum waveguides* – attracted a lot of attention in the last three
decades, both as a source of applications and of interesting mathematical
problems [1,2].

In description of real physical systems the existing models involve various idealizations assuming the confinement realized either by hard walls or by a singular attractive potential of zero ‘width’. This observation motivated recently investigations of situations where the ‘waveguide’ profile consists of regular potential wells or barriers; it was noted that the geometry of the interaction support may itself gave rise to an effective interaction [3–6]. These results inspire many questions that a potential PhD student can address, for instance

- spectral and transport properties of waveguide networks
- potential channels in three dimensions, effects of bending and torsion
- relations between the spectrum and the interaction support geometry
- spectral optimization problems
- resonance effects in such soft waveguides
- the effects of external electric and magnetic fields
- many-body dynamics in this setting

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] P. Exner, H. Kovařík: *Quantum Waveguides*, Springer International,
Heidelberg 2015

[3] P. Exner: *Spectral properties of soft quantum waveguides*, J. Phys.
A: Math. Theor. **53** (2020), 355302

[4] S. Kondej, D. Krejčiřík, J. Kříž: *Soft quantum waveguides with a explicit cut locus*, arXiv:2007.10946

[5] S. Egger, J. Kerner, K. Pankrashkin: *Bound states of a pair of particles
on the half-line with a general interaction potential*, J. Spect. Theory, to
appear; arXiv:1812.06500

[6] S. Egger, J. Kerner, K. Pankrashkin: *Discrete spectrum of Schrödinger
operators with potentials concentrated near conical surfaces*, Lett. Math.
Phys. **110** (2020), 945–968