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arXiv · 0705.2871

On spectral properties of translationally invariant magnetic Schrödinger operators

D. Yafaev
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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 Schrödinger 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 long-time behaviour of solutions $\exp(-i {\bf H} t)f$ of the time dependent Schrödinger 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 one-dimensional free particle but "exits" to $+\infty$ and $-\infty$ might be essentially different.

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D. Yafaev. 2007-05-20. On spectral properties of translationally invariant magnetic Schrödinger operators. https://doi.org/10.1007/s00023-007-0354-4

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