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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
MathSciNet review: 3114848
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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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  • [1] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
  • [2] Akira Iwatsuka, Essential selfadjointness of the Schrödinger operators with magnetic fields diverging at infinity, Publ. Res. Inst. Math. Sci. 26 (1990), no. 5, 841–860. MR 1082319, 10.2977/prims/1195170737
  • [3] Tosio Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241–258. MR 0279626
  • [4] Tosio Kato, Linear evolution equations of “hyperbolic” type. II, J. Math. Soc. Japan 25 (1973), 648–666. MR 0326483
  • [5] Tosio Kato, Remarks on the selfadjointness and related problems for differential operators, Spectral theory of differential operators (Birmingham, Ala., 1981), North-Holland Math. Stud., vol. 55, North-Holland, Amsterdam-New York, 1981, pp. 253–266. MR 640895
  • [6] Tosio Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135–148 (1973). MR 0333833
  • [7] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
  • [8] Herbert Leinfelder and Christian G. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981), no. 1, 1–19. MR 606167, 10.1007/BF01258900
  • [9] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
  • [10] Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945
  • [11] Kenji Yajima, Schrödinger evolution equations with magnetic fields, J. Analyse Math. 56 (1991), 29–76. MR 1243098, 10.1007/BF02820459
  • [12] Kenji Yajima, On time dependent Schrödinger equations, Dispersive nonlinear problems in mathematical physics, Quad. Mat., vol. 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004, pp. 267–329. MR 2231332
  • [13] Kenji Yajima, Schrödinger equations with time-dependent unbounded singular potentials, Rev. Math. Phys. 23 (2011), no. 8, 823–838. MR 2836569, 10.1142/S0129055X11004436

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Additional Information

D. Aiba
Affiliation: Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

K. Yajima
Affiliation: Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

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