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