Laszlo Erdoes
Gaussian Decay of the Magnetic Eigenfunctions
The paper is published: Geom. Funct. Anal. 6, No. 2 (1996) 231-248

MSC:

35Q40 Equations from quantum mechanics

35J10 Schrodinger operator, See also {35Pxx}

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Abstract: We investigate whether the eigenfunctions of the two-dimensional magnetic
Schr\"o\-din\-ger operator have a Gaussian decay of type
$\exp (-C x^2 )$
at infinity (the magnetic field is rotationally symmetric).
We establish this decay
if the energy ($E$) of the eigenfunction is below the
bottom of the essential spectrum ($B$), and if the angular
Fourier components of the external
potential decay exponentially (real analyticity in the angle
variable).
We also demonstrate that almost the same decay is necessary.
The behavior of $C$ in the strong field limit and in the small $(B-E)$
limit is also studied.

Keywords: decay of the eigenfunctions of a magnetic Schroedinger operator, Feynman-Kac formula, Gaussian decay