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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Schrödinger equations with time-dependent strong magnetic fields


Authors: D. Aiba and K. Yajima
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 175-194
MSC (2010): Primary 35J10
Published electronically: March 12, 2014
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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 \vert B(t,x)\vert-C(\vert x\vert^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)$.


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Additional Information

D. Aiba
Affiliation: Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Email: aiba@math.gakushuin.ac.jp

K. Yajima
Affiliation: Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Email: kenji.yajima@gakushuin.ac.jp

DOI: http://dx.doi.org/10.1090/S1061-0022-2014-01284-8
PII: S 1061-0022(2014)01284-8
Keywords: Unitary propagator, Schr\"odinger equation, magnetic field, quantum dynamics, Stummel class, Kato class
Received by editor(s): October 20, 2012
Published electronically: March 12, 2014
Additional Notes: Supported by JSPS grant in aid for scientific research No. 22340029
Dedicated: To the memory of the late Professor Vladimir S. Buslaev
Article copyright: © Copyright 2014 American Mathematical Society