Schr\"odinger equations with time-dependent strong magnetic fields

D. Aiba ; K. Yajima

Algebra i analiz, Tome 25 (2013) no. 2, pp. 37-62.

Résumé

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 $\mathcal{H}=L^2({\mathbb{R}}^d)$ 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 $C_0^{\mathrm{\infty }}({\mathbb{R}}^d)$. 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 $\dot{A}(t,x)$.

Détail

Mots-clés : unitary propagator, Schrödinger equation, magnetic field, quantum dynamics, Stummel class, Kato class.

Bibliographie

[1] Cycon H., Froese R., Kirsch W., Simon B., Schrödinger operators with application to quantum mechanics and global geometry, Springer-Verlag, Berlin, 1987 | MR | Zbl

[2] Iwatsuka A., “Essential self-adjointness of the Schrödinger operators with magnetic fields diverging at infinity”, Publ. Res. Inst. Math. Sci., 26:5 (1990), 841–860 | DOI | MR | Zbl

[3] Kato T., “Linear evolution equations of ‘hyperbolic type’ ”, J. Fac. Sci. Univ. Tokyo, Sec. I, 17 (1970), 214–258 | MR

[4] Kato T., “Linear evolution equations of ‘hyperbolic type’ . II”, J. Math. Soc. Japan, 25 (1973), 684–666 | DOI | MR

[5] Kato T., “Remarks on the essential selfadjointness and related problems for differential operators”, Spectral theory of differential operators, North Holland, Amsterdam, 1981, 253–266 | MR | Zbl

[6] Kato T., “Schrödinger operators with singular potentials”, Israel J. Math., 13 (1973), 135–148 | DOI | MR | Zbl

[7] Lions J.-L., Magenes E., Non-homogeneous boundary value problems and applications, v. I, Grundlehren Math. Wiss., 181, Springer-Verlag, New York–Heidelberg, 1972 | Zbl

[8] Leinfelder H., Simader C., “Schrödinger operators with singular magnetic vector potentials”, Math. Z., 176 (1981), 1–19 | DOI | MR | Zbl

[9] Reed M., Simon B., Methods of modern mathematical physics, v. II, Fourier analysis, selfadjointness, Acad. Press, New York–San Francisco–London, 1975 | MR | Zbl

[10] Yajima K., “Existence of solutions for Schrödinger evolution equations”, Comm. Math. Phys., 110 (1987), 415–426 | DOI | MR | Zbl

[11] Yajima K., “Schrödinger evolution equations with magnetic fields”, J. d'Analyse Math., 56 (1991), 29–76 | DOI | MR | Zbl

[12] Yajima K., “On time dependent Schrödinger equations”, Dispersive nonlinear problems in mathematical physics, Quaderni di Matematica, 15, Seconda Univ. di Napoli, 2005, 267–329 | MR

[13] Yajima K., “Schrödinger equations with time-dependent unbounded singular potentials”, Rev. Math. Phys., 23 (2011), 823–838 | DOI | MR | Zbl