Gaussian decay of the magnetic eigenfunctions
Without magnetic fields it is well-known that the eigenfunctions of Schrödinger operators decay exponentially in the classically forbidden region. In two dimensions with a constant magnetic field and without a potential the decay of the eigenfunctions corresponding to the Landau levels is much stronger, they decay like a Gaussian at infinity. But not much was known in this case with a potential added, except for rotationally invariant potentials. The paper under review fills in this gap. The author investigates the decay of the eigenfunctions of a magnetic Schrödinger operator in two dimensions with a rotationally invariant magnetic field and a potential analytic in the angular variable. For the proof a probabilistic representation of the Schrödinger operator, i.e. a Feynman-Kac formula, in the angular momentum sectors is established. With the help of this representation, it is shown that the eigenfunctions with energy below the essential spectrum exhibit a Gaussian decay at infinity. Due to the probabilistic approach, however, this decay is shown for eigenfunctions with energy below the essential spectrum only. The result has recently been extended to all eigenfunctions with energies in gaps of the essential spectrum, see \textit{S. Nakamura} [Commun. Partial Differ. Equations 21, No. 5-6, 993-1006 (1996)].
decay of the eigenfunctions of a magnetic Schrödinger operator
scientific article; zbMATH DE number 912452