On spectral properties of translationally invariant magnetic Schr\"odinger operators

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 long-time 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 one-dimensional free particle but "exits" to $+\infty$ and $-\infty$ might be essentially different

Scattering by magnetic fields

Consider the scattering amplitude $s(\omega,\omega^\prime;\lambda)$, $\omega,\omega^\prime\in{\Bbb S}^{d-1}$, $\lambda > 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $\omega$ and $\omega^\prime$ away from the diagonal $\omega=\omega^\prime$ but it may be singular on the diagonal. If $d=2$, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac function and of a singular denominator. Such structure is typical for long-range scattering. We refer to this phenomenon as to the long-range Aharonov-Bohm effect. On the contrary, for $d=3$ scattering is essentially of short-range nature although, for example, the magnetic potential $A^{(tr)}(x)$ such that ${\rm curl} A^{(tr)}(x)=B(x)$ and $=0$ decays at infinity as $|x|^{-1}$ only. To be more precise, we show that, up to the diagonal Dirac function (times an explicit function of $\omega$), the scattering amplitude has only a weak singularity in the forward direction $\omega = \omega^\prime$. Our approach relies on a construction in the dimension $d=3$ of a short-range magnetic potential $A (x)$ corresponding to a given short-range magnetic field $B(x)$

The semiclassical limit of eigenfunctions of the Schr\"odinger equation and the Bohr-Sommerfeld quantization condition, revisited

Consider the semiclassical limit, as the Planck constant \hbar\ri 0, of bound states of a quantum particle in a one-dimensional potential well. We justify the semiclassical asymptotics of eigenfunctions and recover the Bohr-Sommerfeld quantization condition