Estimates on spectral gaps for quantum waveguide Schrödinger operators

• Applicant: Professor Dr. Ivan Veselic
• Subject Area: Mathematics
• Term: from 2006 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 27091790

The aim of the project is to study spectral properties of Schrödinger operators which describe quantum waveguides and so-called leaky quantum wires. The strong interest of studying this kind of models comes from mesoscopic physics. Quantum waveguides describe an electron moving in a semiconductor structure producing potential barriers which can be modelled as hard walls. Leaky quantum wires describe the electron movement between two different semiconductor materials giving rise to a finite potential jump.Operators describing quantum waveguides are given by the Laplacian restricted to a tube in three dimensional Euclidean space or a strip in two dimensional Euclidean space. It is known that even for infinitely long tubes/strips non-trivial curvature can induce bound states. In this situation a natural question arises: can one give estimates on the distance between consecutive eigenvalues. As usual, lower bounds on the spectral gaps are more challenging to derive than upper ones. The aim of the project is to derive such lower bounds, in particular for the low lying eigenvalues.The same question on lower bounds for eigenvalue splittings can be posed for a Schrödinger operator which models leaky quantum wires. The operator consists of the negative Laplacian corresponding to the kinetic energy and a singular interaction as the potential term. The singular potential is attractive and supported on an interface manifold. Under mild assumptions this Hamiltonian has bound states. The aim of the project is again to derive lower bounds for the distance of consecutive eigenvalues.

• DFG Programme: Research Grants