Schrödinger equations with time-dependent strong magnetic fields

Abstract:

Time dependent $d$-dimensional Schrödinger equations $i\partial _t u = H(t)u$, $H(t)=-(\partial _x-iA(t,x))^2+ V(t,x)$ are considered in the Hilbert space $\mathcal {G}=L^2(\mathbb {R}^d)$ 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 $\theta <1$ and $C>0$. It is shown that such equations uniquely generate unitary propagators in $\mathcal {G}$ under suitable conditions on the size and singularities of the time derivatives of the potentials $\dot V(t,x)$ and $\dot A(t,x)$.