RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (364 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Language:

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
\Bibitem{AibYaj13} \by D.~Aiba, K.~Yajima \paper Schr\"odinger equations with time-dependent strong magnetic fields \jour Algebra i Analiz \yr 2013 \vol 25 \issue 2 \pages 37--62 \mathnet{http://mi.mathnet.ru/aa1322} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3114848} \zmath{https://zbmath.org/?q=an:1304.35629} \elib{http://elibrary.ru/item.asp?id=20730196} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000343074000002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84924406384} 

• http://mi.mathnet.ru/eng/aa1322
• http://mi.mathnet.ru/eng/aa/v25/i2/p37

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. K. Yajima, “Existence and regularity of propagators for multi-particle Schrödinger equations in external fields”, Commun. Math. Phys., 347:1 (2016), 103–126
3. A. Michelangeli, A. Olgiati, “Gross–Pitaevskii non-linear dynamics for pseudo-spinor condensates”, J. Nonlinear Math. Phys., 24:3 (2017), 426–464
•  Number of views: This page: 240 Full text: 52 References: 34 First page: 13