Vladimir Kondratiev, Vladimir Maz'ya, Mikhail Shubin
Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators
Preprint series: ESI preprints

MSC:

35J10 Schrodinger operator, See also {35Pxx}

58G25 Spectral problems; spectral geometry; scattering theory, See also {35Pxx}

35P15 Estimation of eigenvalues, upper and lower bounds

47F05 Partial differential operators, See also {35Pxx, 58G05}

Abstract: We establish necessary and sufficient conditions
for the discreteness of spectrum and strict positivity
of magnetic Schr\"odinger operators with a positive scalar potential.
They are expressed in terms of Wiener's capacity and
the local energy of the magnetic field.
The conditions for the discreteness of spectrum depend, in particular, on
a functional parameter which is a decreasing function of one variable
whose argument is the normalized local energy of the magnetic field.
This function enters the negligibility condition of sets for the
scalar potential.
We give a description for the range of all admissible functions
which is precise in a certain sense.

In case when there is no magnetic field, our results extend
the discreteness of spectrum and positivity criteria by A.~Molchanov (1953)
and V.~Maz'ya (1973).