Geodesic
Schr\"odinger equations with time-dependent strong magnetic fields
Algebra i analiz, Tome 25 (2013) no. 2, pp. 37-62.

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Time dependent $d$-dimensional Schrödinger equations $i\partial_tu=H(t)u$, $H(t)=-(\partial_x-iA(t,x))^2+V(t,x)$ are considered in the Hilbert space $\mathcal H=L^2(\mathbb R^d)$ of square integrable functions. $V(t,x)$ and $A(t,x)$ are assumed to be almost critically singular with respect to the spatial variables $x\in\mathbb R^d$ both locally and at infinity for the operator $H(t)$ to be essentially selfadjoint on $C_0^\infty(\mathbb R^d)$. In particular, when the magnetic fields $B(t,x)$ produced by $A(t,x)$ are very strong at infinity, $V(t,x)$ can explode to the negative infinity like $-\theta|B(t,x)|-C(|x|^2+1)$ for some $\theta1$ and $C>0$. It is shown that such equations uniquely generate unitary propagators in $\mathcal H$ under suitable conditions on the size and singularities of the time derivatives of the potentials $\dot V(t,x)$ and $\dot A(t,x)$.
Keywords: unitary propagator, Schrödinger equation, magnetic field, quantum dynamics, Stummel class, Kato class. 
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D. Aiba; K. Yajima. Schr\"odinger equations with time-dependent strong magnetic fields. Algebra i analiz, Tome 25 (2013) no. 2, pp. 37-62. http://geodesic.mathdoc.fr/item/AA_2013_25_2_a1/

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