Separation of different forms of the Dirac operator in Hilbert spaces. (English) Zbl 1080.34569
Summary: We study the separation property of two different forms of the Dirac differential operator
(i) \(Lu(x)=i^{-1}\alpha.\text{grad} \,u(x)+ \beta(x) u(x),x\in \mathbb{R}^3\) in the Hilbert space \(H_1=L_2(\mathbb{R}^3)^4\) and
(ii) \(Gu(x)=i^{-1} B\frac{d}{dx}u(x)+v(x)u(x),x\in \mathbb{R}\) in the Hilbert space \(H_2=L_2(\mathbb{R})^\ell\).
(i) \(Lu(x)=i^{-1}\alpha.\text{grad} \,u(x)+ \beta(x) u(x),x\in \mathbb{R}^3\) in the Hilbert space \(H_1=L_2(\mathbb{R}^3)^4\) and
(ii) \(Gu(x)=i^{-1} B\frac{d}{dx}u(x)+v(x)u(x),x\in \mathbb{R}\) in the Hilbert space \(H_2=L_2(\mathbb{R})^\ell\).
MSC:
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 35Q40 | PDEs in connection with quantum mechanics |
