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This article is cited in 5 scientific papers (total in 5 papers)
Research Papers
Schrödinger equations with time-dependent strong magnetic fields
D. Aiba, K. Yajima Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Abstract:
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 $\theta<1$ 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.
Received: 20.10.2012
Citation:
D. Aiba, K. Yajima, “Schrödinger equations with time-dependent strong magnetic fields”, Algebra i Analiz, 25:2 (2013), 37–62; St. Petersburg Math. J., 25:2 (2014), 175–194
Linking options:
https://www.mathnet.ru/eng/aa1322 https://www.mathnet.ru/eng/aa/v25/i2/p37
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Abstract page: | 391 | Full-text PDF : | 97 | References: | 64 | First page: | 13 |
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