Schr\"odinger equations with time-dependent strong magnetic fields
Schr\"odinger equations with time-dependent strong magnetic fields
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具有随时间变化的强磁场的 Schr"odinger 方程
DOI：
10.1090/s1061-0022-2014-01284-8
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
K. Yajima
中科院分区：
文献类型：
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作者：
D. Aiba;K. Yajima
考虑平方可积函数的Hilbert空间上的d维含时Schr“odinger方程.我们假设磁势和标量势关于空间变量在局部和无穷远处都是几乎临界奇异的，使得固定时间Schr“odinger算子H（t）在紧支撑光滑函数上本质上是自伴的.特别地，如果磁场B（t，x）在无穷远处非常强，则标量势可以比二次函数更快地爆炸到负无穷远处。我们表明，方程唯一生成酉传播在适当的条件下的大小和奇异性的时间导数的潜力。基本工具是加藤的抽象理论的发展方程，岩冢的身份重写H（t）的椭圆微分算子中，B（t，x）出现明确的，和一个新的抗磁一样的不等式。
We consider d-dimensional time dependent Schr\"odinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and at infinity for the fixed time Schr\"odinger operator H(t) to be essentially self-adjoint on the compactly supported smooth functions. In particular, if magnetic field B(t,x) is very strong at infinity, the scalar potential can explode to negative infinity faster than quadratic functions. We show that equations uniquely generate unitary propagators under suitable conditions on the size and singularities of time derivatives of potentials. Basic tools are Kato's abstract theory for evolution equations, Iwatsuka's identity which rewrites H(t) to an elliptic differential operator in which B(t,x) appears explicitly, and a new diamagnetic like inequality.