Spectral properties of soft quantum waveguides

Pavel Exner (Czech Academy of Sciences, Prague)

Nov 07. 2022, 09:30 — 10:15

The topic of this talk are `soft' quantum waveguides described by Schr\"odin\-ger operators with an attractive potential in the form of a channel of a fixed profile built along a smooth curve in $\mathbb{R}^\nu$. In the case when $\nu=2$ and the curve is infinite and not straight, but asymptotically straight in a suitable sense, we derive using Birman-Schwinger principle a sufficient condition for the discrete spectrum of such an operator to be nonempty; this happens, in particular, when the potential well defining the channel profile is deep and narrow enough. Under a restriction on the waveguide torsion, this results extends to curves in $\mathbb{R}^3$. We also address the question about ground state opti\-mization in the situation when the generating curve in $\mathbb{R}^2$ has the shape of a loop without self-intersections. Some related results and problems are also mentioned.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Spectral Theory of Differential Operators in Quantum Theory (Workshop)
Jussi Behrndt (TU Graz)
Fritz Gesztesy (Baylor U, Waco)
Ari Laptev (Imperial College London)
Christiane Tretter (U Bern)