arXiv subjects
Publications and source records attributed to Daisuke Aiba.
We consider the massless Dirac operator $H=α\cdot D+Q(x)$ on the Hilbert space $L^{2}(\mathbb{R}^{3},\mathbb{C}^{4})$, where $Q(x)$ is a $4\times4$ Hermitian matrix valued function which suitably decays at infinity. We show that the the zero resonance is absent for $H$, extending recent results of Saitō-Umeda and Zhong-Gao.
We consider d-dimensional time dependent Schrödinger 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 Schrödinger 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.