Gauge optimization and spectral properties of magnetic Schrödinger operators

Kondratiev, Vladimir; Maz'ya, Vladimir; Shubin, Mikhail

Gauge optimization and spectral properties of magnetic Schrödinger operators

Authors

Kondratiev, Vladimir

Maz'ya, Vladimir

Shubin, Mikhail

Description

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They extend earlier results by Maz'ya and Shubin (2005), which were obtained in case when there is no magnetic field. We also derive two-sided estimates for the bottoms of spectrum and essential spectrum, extending results by Maz'ya and Otelbaev (1977). The main idea is to optimize the gauge of the magnetic field, thus reducing the quadratic form to one without magnetic field (but with an appropriately adjusted scalar potential).
20 pages