 07123 D. Yafaev
 On Spectral Properties of Translationally Invariant Magnetic Schr\"odinger Operators
(83K, latex)
May 20, 07

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Abstract. We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction.
Our goal is to study basic spectral
properties of the Schr\"odinger operator ${\bf H}$ with such a potential. In particular, we show that the spectrum of ${\bf H}$ is
absolutely continuous and we find its location. Then we study the
longtime behaviour of solutions $\exp(i {\bf H} t)f$ of the time dependent Schr\"odinger equation.
It turnes out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to a evolution of a onedimensional free particle but ``exits" to $+\infty$ and $\infty$ might be essentially different.
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