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Algebra i Analiz, 2013, Volume 25, Issue 2, Pages 37–62 (Mi aa1322)  

This article is cited in 3 scientific papers (total in 3 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.

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English version:
St. Petersburg Mathematical Journal, 2014, 25:2, 175–194

Bibliographic databases:

Received: 20.10.2012
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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

Citation in format AMSBIB
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\by D.~Aiba, K.~Yajima
\paper Schr\"odinger equations with time-dependent strong magnetic fields
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\yr 2013
\vol 25
\issue 2
\pages 37--62
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\transl
\jour St. Petersburg Math. J.
\yr 2014
\vol 25
\issue 2
\pages 175--194
\crossref{https://doi.org/10.1090/S1061-0022-2014-01284-8}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Michelangeli, “Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields”, Nonlinearity, 28:8 (2015), 2743–2765  crossref  mathscinet  zmath  isi  scopus
    2. K. Yajima, “Existence and regularity of propagators for multi-particle Schrödinger equations in external fields”, Commun. Math. Phys., 347:1 (2016), 103–126  crossref  mathscinet  zmath  isi  scopus
    3. A. Michelangeli, A. Olgiati, “Gross–Pitaevskii non-linear dynamics for pseudo-spinor condensates”, J. Nonlinear Math. Phys., 24:3 (2017), 426–464  crossref  mathscinet  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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