Stability of the magnetic Schrödinger operator in a waveguide

Details Stability of the magnetic Schrödinger operator in a waveguide Stability of the magnetic Schrödinger operator in a waveguide

2004-04-28

arxiv.org/abs/math-ph/0404069v1

Commun. in Part. Diff. Eq. 30 (2005) 539-565

Physics

Mathematical Physics

Scientific paper

10.1081/PDE-200050113

The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schr\"odinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own.

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