Geometric perturbations of soft quantum waveguides

Mathphys Analysis Seminar

Soft quantum waveguides are quantum mechanical systems in which a
particle is confined by an attractive potential in the vicinity of a curve,
rather than by a hard boundary or singular interaction. I will discuss two
geometric perturbations of such systems. The first concerns an array of
potential wells in dimensions two and higher, where the axial symmetry is
discrete. Shifting finitely many wells along the axis, even a single one and by
an arbitrarily small amount, produces a bound state below the essential
spectrum. The second concerns a twisted soft waveguide in three dimensions,
for which I will present a Hardy inequality. It shows that the Hamiltonian is
subcritical, so that weak attractive perturbations do not bind.